Method of reducing multipole content in a conductor assembly during manufacture

ABSTRACT

A method for manufacture of a conductor assembly. The assembly is of the type which, when conducting current, generates a magnetic field or in which, in the presence of a changing magnetic field, a voltage is induced. In an example embodiment one or more first coil rows are formed. The assembly has multiple coil rows about an axis with outer coil rows formed about inner coil rows. A determination is made of deviations from specifications associated with the formed one or more first coil rows. One or more deviations correspond to a magnitude of a multipole field component which departs from a field specification. Based on the deviations, one or more wiring patterns are generated for one or more second coil rows to be formed about the one or more first coil rows. The one or more second coil rows are formed in the assembly. The magnitude of each multipole field component that departs from the field specification is offset.

RELATED APPLICATION

This application claims priority to provisional patent application U.S.60/976,985 filed 2 Oct. 2007 which is incorporated herein by referencein the entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

The United States Government may have certain rights in this inventionpursuant to U.S. Government Contract Number DE-FG02-06ER84492 awarded bythe United States Department of Energy.

FIELD OF THE INVENTION

This invention relates to electromagnetic systems which generatemagnetic fields. More particularly, the invention relates to systems ofthe type including conductor assemblies which, when conducting current,generate a magnetic field or which, in the presence of a changingmagnetic field, generate or transform voltages.

It is of continued importance across many sectors of the world economy(e.g., R&D, and medical applications) to achieve improved performance inmagnetic conductor assemblies. Development of new and improvedcommercial applications is dependent on an ability to create large anduniform magnetic fields. Advancements are also needed in numerousperformance and reliability factors to realize commercially usefulembodiments in medical, industrial and commercial applications. Forexample, it is desirable to make charged particle therapy cancertreatment (e.g., proton and carbon therapy) more available to patients,but these systems require cyclotrons and very large magnets to steerbeams of high energy charged particles. System size and cost severelylimit the availability of these applications. Currently, the gantriesused for proton therapy treatment rooms may extend multiple stories inheight and weigh over one hundred tons. One impediment to furtherdeployment of these and other charged particle beam systems is the sizeand cost of the beam acceleration and focusing equipment.

In the long term, for charged particle therapy and certain other highmagnetic field applications, it is likely that superconducting magnetswill be preferred over resistive magnets. Generally, superconductingmagnets offer very stable and high field strengths and can besubstantially smaller in size than resistive magnets. Moreover, thepower demands of superconducting magnets are very low. However, theopportunity to provide superconducting magnets in new applications maybe compromised because of the well-known quenching phenomenon. When thesuperconducting material undergoes an unexpected and rapid transition toa normal, non-superconducting state this can result in rapid formationof a high temperature hot spot which can destroy a magnet. Designs whichimprove reliability have been costly. Cost is a major constraint togreater commercialization of conventional superconducting magnettechnologies which rely on saddle or racetrack coils. Moreover, for agiven set of operating conditions, significant design efforts must beemployed to achieve requirements of field uniformity and to assure thatquenching does not occur during normal system use.

Whether future systems employ resistive or superconductive windings, aneed will remain to improve design efficiency, reliability and fieldquality. In order to deploy carbon-based systems for charged particlecancer treatment, the use of superconducting magnets may be imperativein order to meet the bending requirements of the high energy carbonbeam. Coil segments used to bend beams are very complex and must be verystable in order to implement a curved trajectory. Further, it is verydifficult to apply conventional geometries, e.g., saddle coil and racetrack configurations, to curvilinear applications and still meetrequirements for field configurations.

At the same time, it is necessary to provide these systems at lowercosts in order to encourage wider uses that benefit society. By way ofillustration, mechanical structures required to assure stabilization ofconductor windings in the presence of large fields are effective, butthey are also a significant factor in overall weight and system cost.There is a continuing need to build magnet systems which are moreefficient, more robust and more reliable. As one example, with rotatingmachinery being subject to wear under conditions of continued use, thereare needs to provide costly maintenance and repair. Design improvementswhich substantially reduce these life cycle costs and the overallaffordability of high field systems can accelerate deployment of usefulsystems that require generation of large magnetic fields.

SUMMARY OF THE INVENTION

In a series of embodiments according to the invention, a method isprovided for manufacture of a conductor assembly having multiple coilrows about an axis with outer coil rows formed about inner coil rows.The assembly is of the type which, when conducting current, generates amagnetic field or in which, in the presence of a changing magneticfield, a voltage is induced. In an example embodiment one or more firstcoil rows are formed. A determination is made of deviations fromspecifications associated with the formed one or more first coil rows,with one or more deviations each corresponding to a magnitude of amultipole field component which departs from a field specification.Based on the one or more deviations, one or more wiring patterns aregenerated for one or more second coil rows to be formed about the one ormore first coil rows. Accordingly, the magnitude of each multipole fieldcomponent that departs from the field specification is offset. The oneor more second coil rows are formed in the assembly.

In another example of a method of manufacture according to theinvention, measurement is made of a deviation of a multipole order froma multipole field specification in one or multiple fully formed coilrows. Offsetting corrections are introduced during fabrication of anoverlying coil row to reduce deviation from the multipole fieldspecification in the completed assembly.

In still another example, a fabrication method suppresses generation ofunwanted multipole field orders in a conductor assembly comprisingmultiple coil rows about an axis. Outer coil rows are formed about innercoil rows, and inner and outer ones of the coil rows comprise conductorformed about the axis in a helical pattern. After forming one or morefirst inner coil rows, one or more outer coil rows are formed. Theprocess of forming the one or more outer coil rows includes introducinga modulation into the one or more outer coil rows to reduce themagnitude of one or more multipole field components relative to a mainfield component.

Many applications of magnets for charged particle beam optics canbenefit from magnets that are curved. According to embodiments of theinvention, this geometry can be implemented with pure multipole fields,i.e., predominantly one multipole order, by constructing double helixcoil designs along curvilinear axes. A uniform magnetic dipole field canbe applied to bend a charged particle beam trajectory into an arc. Amagnet coil can be designed about a curved axis in order to avoidinterference between the particle beam and the inner aperture of themagnet coil, e.g., a stainless steel vacuum tube.

Simply transforming a double helix coil from a straight configuration,which generates a dipole field along a straight axis, into an arc ofvariable and arbitrary curvature generally leads to the introduction ofhigher-order multipole field components. Intuitively, this can beunderstood in the following way. When the straight pattern in FIG. 2 isbent around the Z-axis, conductors on the inner arc of the bend areforced closer together and the conductors on the outer arc of the bendare spread further apart. The resulting difference in wire spacing isdue to the fact that the circumference on the inner arc is smaller thanon the outer arc while the number of turns is the same. The consequentvariation in wire spacing leads to an increased field near the wireportions positioned more closely together and a lesser field where thespacing is greater, this leading to a gradient in the field, which willbe mainly a quadrupole component.

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1A and 1B are, respectively, perspective and elevation views ofthree-dimensional space curves illustrating a simple prior art spiralpattern;

FIG. 2 is a perspective view of a prior art coil having a regularhelical geometry as used to form prior art double helix coil pairssuitable for generating a dipole field;

FIG. 3 is a perspective view of a prior art coil pattern used to formprior art double helix coil pairs suitable for generating a quadrupolefield;

FIG. 4 is a perspective view of a prior art coil pair wherein the twocoil patterns have opposite tilt angles relative to a plane;

FIG. 5 is an unrolled view of the quadrupole coil pattern shown in FIG.3;

FIG. 6 is an unrolled view of a wiring pattern comprising multiplemultipole components according to the prior art;

FIG. 7 is an unrolled view of a wiring pattern comprising multiplemultipole components according to an embodiment of the invention;

FIG. 8 is a perspective view of a magnet having a curved apertureconstructed according to an embodiment of the invention;

FIGS. 9A and 9B illustrate relationships corresponding to coordinatetransformation between straight and curved helical coil rows;

FIG. 10 illustrates an iterative process according to a methodology fordetermining values of geometric parameters for construction of thecurved helical coil row of FIG. 9 according to multipole componentspecifications;

FIG. 11 is a perspective view of a magnet design used in practicing amethod according to the invention;

FIGS. 12A, 12 B and 12C provide, respectively, a perspective view, aview in cross section about an axis of symmetry and a view in crosssection along the axis, of a magnet according to an embodiment of theinvention;

FIGS. 13A-13E are elevation views and FIGS. 13F-13G are views in crosssection illustrating sequences of fabrication and features offabrication according to the invention;

FIG. 14 illustrates a double helix pair of coil rows useful forformation of a combined function magnet according to the invention;

FIGS. 15A-15C illustrate in elevation views features of a fabricationprocess according to the invention, while FIG. 15D illustrates a view incross section of a channel during fabrication;

FIGS. 16A and 16B illustrate sextupole field magnitudes along a centralaxis of the coil assembly shown in FIG. 11;

FIGS. 17A and 17B illustrate a coil assembly according to anotherembodiment of the invention in perspective and elevation views,respectively;

FIGS. 18 and 19 illustrate alternate embodiments of an iterative processaccording to a methodology for determining values of geometricparameters for construction of coil rows according to multipolecomponent specifications;

FIG. 20A provides an unrolled view of a coil roll pattern fabricatedaccording to a Direct Helix manufacturing process;

FIG. 20B is a partial unrolled view of conductive strips shown in FIG.20A;

FIG. 21 is a view in cross section of conductive strips in the coil rowof FIG. 20;

FIG. 22 is a perspective view illustrating features of a Direct Helixcoil; and

FIG. 23 is a schematic view of a magnet assembly comprising coilsections of varied design according to the invention.

Like reference numbers are used throughout the figures to denote likecomponents. Numerous components are illustrated schematically, it beingunderstood that various details, connections and components of anapparent nature are not shown in order to emphasize features of theinvention. Various features shown in the figures are not shown to scalein order to emphasize features of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Before describing in detail the particular methods and apparatusesrelated to embodiments of the invention, it is noted that the presentinvention resides primarily in a novel and non-obvious combination ofcomponents and process steps. So as not to obscure the disclosure withdetails that will be readily apparent to those skilled in the art,certain conventional components and steps have been omitted or presentedwith lesser detail, while the drawings and the specification describe ingreater detail other elements and steps pertinent to understanding theinvention. Further, the following embodiments do not define limits as tostructure or method according to the invention, but provide exampleswhich include features that are permissive rather than mandatory andillustrative rather than exhaustive.

As used herein, the terms coil, spiral, helix and helical include butare not limited to regular geometric patterns. In addition, the termscoil, spiral and helix include configurations wherein a width (e.g.,along the axial direction) or a thickness (e.g., along a radialdirection or transverse to the axial direction) may vary. Contemplatedembodiments include variations which depart substantially from regulargeometries and which therefore may not be simply described in closedform. Numerical solutions, proximate as they may be, can be applied tomodel and design wiring configurations which may then be constructedaccordingly to a desired level of precision. Further, terms such aswinding, helical winding, wiring pattern and coil configuration asapplied to physical embodiments formed of various conductor and/orinsulative materials, are used without regard to how the materials areformed in place. That is, although it is conventional to physically winda strand of conductor in the configuration of a spiral, the foregoingterms as used herein refer to the resulting configuration and not themethodology used to form the pattern. So, for example, a coil or windingmay be formed from a cylindrical body by removal of body material, thisresulting in a shape that corresponds to a spiral winding. In addition,the void resulting from the removal of material may also correspond to aspiral shape.

With coils helically-wound about an axis to produce magnetic fieldcomponents transverse to the axis, cancellation of axial fieldcomponents can be effected by the formation coils in concentricallypositioned pairs having opposite tilt angles, this sometimes resultingin a high quality transverse field, e.g., a uniform dipole withessentially no higher order components. See, for example, Goodzeit etal., “The Double-Helix Dipole—A Novel Approach to Accelerator MagnetDesign”, IEEE Transactions on Applied Superconductivity, Vol. 13, No. 2,June 2003, pp. 1365-1368, which describes analytics for a double helixmagnet geometry. See, also, U.S. Pat. No. 6,921,042 incorporated hereinby reference.

For helically wound conductors and other magnet geometries, some ofthese being racetrack and saddle configurations, placement of conductorhas been problematic for multiple reasons. In conventional racetrack andsaddle configurations, based on circular shaped-cable, the position ofeach wire turn has depended on the position of a previous wire turn.Such windings typically build on one another with a second row of turnsbeing tightly wound over a previously wound row of turns. The windingsare often generated with assistance of tooling that assures consistencyas turns in each row are wound tightly against one another and as turnsin consecutive rows are created one over the other. This tight stackingof turns has provided a means to stabilize the conductor. Further, thistype of configuration often results in contact between turns in the samerow as well as between turns in adjoining rows, and has requiredinsulative coating on the conductor surface so that portions of theconductor coming into contact with other portions of the conductor areinsulated from one another. To assure stability of the winding underhigh field conditions the turns are commonly bonded to one another with,for example, an adhesive.

In these prior systems the position and stability of the conductor hasdepended on the positioning of each conductor turn against anotherconductor turn and the ability to maintain the conductor in a staticposition during manufacture, assembly, and operation, i.e, under typicalthermal cycling and high Lorentz forces acting during coil excitation.While the required tight nesting of turns of insulated wire withoutintervening layers can stabilize the conductor, the design of the wiringpattern has been limited and, thus, variation in design of the fieldpattern has also been limited. As shown in the illustrated embodiments,it is now possible to more fully utilize other wiring patterns, withoutcompromising reliability, by separating all of the rows of conductorsegments with intervening insulative layers and pre-defining the wiringpatterns with channels formed in the insulative layers. Such techniquesare more fully described in co-pending U.S. application Ser. No.12/061,813 “Wiring Assembly and Method of Forming A Channel In A WiringAssembly For Receiving Conductor” filed Apr. 3, 2008 assigned to theassignee of the present invention and now incorporated herein byreference.

Formation of channels into which the conductor is inserted providesprecise conductor positioning and stabilization while also isolatingportions of the conductor from other portions of the conductor. Thechannel profile is not limited to accommodating round wire or cables.Other conductors having square or rectangular shapes in cross section,or tape, can be used in conjunction with channels. The channel may beconfigured to match the cross sectional shape of the conductor. Theconductor pattern and the corresponding channel path can be formed in arelatively tight helical configuration wherein h, the advance per turnin an axial direction, is so small that portions of the conductor inadjacent turns come very close or into contact with one another. Inembodiments where contact between adjacent portions of conductor turnsis a concern, the conductor has an insulative coating.

When the channels accommodate square or rectangular cross sectionalshapes of conductor, including tape, to minimize deformation inconductor, e.g., twisting, a helical channel can be formed at a variableangle with respect to a central axis or reference surface. In suchembodiments, the resulting field will differ from that which isgenerated for a conventional conductor of circular cross sectionalshape. A channel for a circular shaped conductor will not follow thesame path as a channel formed at such variable angle to accommodate arectangular shaped conductor without shape deformation.

The term “conductor” as used herein refers to a string-like piece orfilament of relatively rigid or flexible material, commonly referred toas cable or wire, being of the type comprising either a singleconductive strand or multiple ones of such strands grouped together asone functional conductive path. The term multi-strand conductor refersto such a conductor formed as a single identifiable unit and composed ofmultiple conductive strands which may be twisted, woven, braided orintertwined with one another to form an identifiable single unit ofwire. Multi-strand conductor may take the form of conductor thatembodies a circular or a non-circular cross section.

The term cross section refers to a section of a feature, e.g., of aconductor or an aperture or a coil, taken along a plane which istransverse to a definable axis through which the feature extends. If thecoil row axis is curvilinear about a point of interest on the axis, theplane along which the cross section is taken is understood to betransverse to the direction of a vector which is tangent to thedirection of the axis at the point of interest.

A simple prior art spiral pattern in three-dimensional space, shown inthe perspective view of FIG. 1A and the elevation view of FIG. 1B, isgenerated in accord with the relationships 1A, 1B and 1C:X(θ)=[h/(2*π)]θ  1AY(θ)=R cos(θ)  1BZ(θ)=R sin(θ)  1Cwherein the X coordinate is along a longitudinal direction parallel withan axis of symmetry and the Y and Z coordinates are along directionstransverse to the axis of symmetry and orthogonal to one another. θ isthe azimuthal angle measured in a Y-Z plane transverse to the X-axis.The parameter h defines the advance per turn in the X direction. R isthe radius of the aperture of the winding pattern. That is, forembodiments having a regular shape, R corresponds to a radial distancefrom an axis of symmetry to a point on the curve, and the aperture isthe volume within the shape formed by the helical pattern.

FIGS. 2 and 3 are exemplary three-dimensional space curves illustratingfeatures of prior art coils found in double helix coil pairs. Forpurposes of clarity, FIGS. 2 and 3 each illustrate a single coil row.These rows correspond to regular helical geometries generated in accordwith the relationships 2A, 2B and 2C:X(θ)=[h/(2*π)]θ+A _(n) sin(nθ)  2AY(θ)=R cos(θ)  2BZ(θ)=R sin(θ).  2C

The curve for n=1 is shown in the perspective view of FIG. 2. The curvefor n=2 is shown in the perspective view of FIG. 3.

The term A_(n) sin(nθ), in the X(θ) equation, imparts a positive or anegative tilt to each of the turns relative to the Y-Z plane, inproportion to the magnitude and sign of the term A_(n). According to thevalue of n, the term A_(n) sin(nθ) also introduces a modulation, i.e., asinusoidal variation, in each 360 degree turn of the curve about theaxis. For n=1, an ellipsoidal shape is imparted to each turn as shown inFIG. 2. The more complex pattern shown in FIG. 3, having a higher ordersinusoidal component (n=2), is suitable for generating a quadrupolefield. For higher values of n, still higher frequency sinusoidalcomponents modulate the shape of each turn.

As can be seen from FIG. 2, with addition of the A_(n) sin(nθ) term andwith n=1, the turns are tilted relative to the YZ-plane. This results ina significant component of current flow in the axial direction. Atransverse magnetic field is therefore generated together with an axialfield component. With incorporation of a second layer of turns (as shownin FIG. 4, again with n=1), and with the two patterns having oppositetilt angles relative to the YZ-plane (by providing the terms A_(n) ineach of the two coils with opposite signs), it is possible to generate asubstantially pure transverse field and practically eliminate the axialfield component. This and other pairs of coil patterns having oppositetilts, i.e., for differing values of n, are referred to in theliterature as double-helix windings.

Still, more generally, for several embodiments of the invention, athree-dimensional space curve may be generated in accord with theequations 3A, 3B and 3C:X(θ)=[h/(2*π)]θ+ΣA _(n) sin(nθ+φ _(n))  3AY(θ)=R cos(θ)  3BZ(θ)=R sin(θ)  3Cwherein A_(n) determines the amplitudes in equation 3A, and φ_(n)determines phase shifts between the sinusoidal components. R determinesthe radius of the winding pattern, which is measured from the axis ofthe cylindrically shaped coil and θ is the azimuth angle. In thiscontext the term coil and the adjective helix refer to a variety ofspiral-like shapes which can result from the aforedescribed function,understanding that other trigonometric or numerical expressions may beused to define the channel path and the conductor path. The individualor combined content of the fields corresponding to one or more values ofn are generally referred to as multipole moments. Field componentsgenerated from a double-helix winding configuration, and correspondingto different values of n according to equation 3 are substantially orentirely orthogonal with one another.

An individual layer of a double-helix coil simultaneously generatestransverse and axial magnetic fields. Transverse in this context denotesmagnetic fields having Y and Z components. In most applications thecurrent directions in individual layers of double-helix coils are chosenin such a way that the transverse magnetic fields of layers add up,while the axial fields are canceled to a high degree. It is thereforecustomary to describe the magnetic field by two dimensional multipolesin the transverse plane. If the field changes along the X-direction,e.g. as is the case near the coil ends, a two dimensional multipoleexpansion can still be used to describe the field, and the multipolecontents for different axial positions are determined. In accord withequation 3A, the multipole field components that can be generated withthe resulting coil pattern correspond to the values of n for which eachA_(n) is nonzero in equation 3A.

In a long winding configuration, where coil end effects can beneglected, the pattern for n=1 will generate an essentially pure dipolefield having no higher order components. Similarly, a quadrupole pattern(n=2), a sextupole pattern (n=3) and other higher order patternsgenerate pure fields with a multipole order defined by the value of n.

Theoretically, magnetic fields of almost arbitrary shape and quality canbe generated in accord with the above mathematics. However, constructionof coils for generating fields with higher multipole order (n>1) orfields containing more than one multipole order, e.g., superimposeddipole plus quadrupole fields, is limited by geometrical constraints,such as requiring a minimum spacing between conductors to avoidconductor impingement. The conductor spacing in a coil is controlled bythe term, h, in equation 3A. For increasing values of h the conductorsare spaced further apart along the X-direction. The minimum conductorspacing corresponds to when adjacent conductors just touch each other.Any further decrease in conductor spacing would lead to interferencebetween neighboring conductors.

FIG. 5 presents a 360 degree view of the quadrupole coil pattern shownin FIG. 3. This and other 360 views of coil patterns shown in FIGS. 6and 7 are transforms from views of three dimensional contours such asthe cylindrical-like configuration of FIG. 3, to views in a plane,referred to herein as “unrolled” views. That is, these views aregenerated as though the three dimensional shaped surface is cut open andlaid along a plane to provide a two dimensional or plan view in whichthe abscissa represents the arc length over the cylinder surface and theordinate represents the axial direction.

The minimum required conductor spacing can be illustrated in an unrolledview of the winding pattern, where the X-coordinate is plotted againstthe circumference U, which is given by the radius R times the azimuthangle, θ). As shown in FIG. 5, the local slope of the conductordirection is dX/dU=tan(α) where α is the slope angle in the unrolledview, which depends on the azimuth angle θ. The minimum possible wirespacing without impingement is given as follows by equations 4A and 4B:tan(α)=dX/dU=(1/R)(dX/dθ)  4Aminimum spacing=d/cos(α_(max)),  4Bwhere α_(max) is the maximum slope angle incurred along the trajectory.As can be seen from equation 4B, the minimum spacing is determined bythe largest slope angle α in the coil winding. See FIG. 5 for anillustration of the slope angle α. Also, as illustrated in FIGS. 5, 6and 7, the illustrated wiring patterns are a continuous series ofsegments 2. Along first portions 4 of the segments, individual segmentsare relatively straight and along second portions 6 of the segments thesegments follow a contour having a definable radius of curvature.

Larger slope angles require larger conductor spacings in a windingpattern and thereby lower the achievable magnetic field strength of theresulting coil configuration. This is because fewer conductor turns canbe applied per unit distance along the X axis. Many applications requirerelatively high field strengths and it may be desirable to achieve theminimum possible conductor spacing (e.g., with the conductor surfaceshaving an insulative coating enabling surfaces to touch one another) asdefined in equation 4B. Since the higher-order multipole windingconfigurations have more sinusoidal oscillations per conductor turn (seeequation 3A), the slope angles α generally increase with increasingmultipole order content.

The minimum possible conductor spacing in combined function magnets isalso affected by the phase angles φ_(n). See equation 3A. Qualitativelythis can be understood for superimposed dipole and quadrupole fieldsaccording toX(θ)=[h/(2*π)]θ+A ₁ sin(θ)+A ₂ sin(2θ+Δφ)  5AFor Δφ=0, minima and maxima of the dipole component coincide with minimaand maxima of the quadrupole component, while for a Δφ≠0 the peak valuesof the component sinusoidal functions are displaced. For example,referring to Equation 3A, with φ_(i) not equal to φ_(j) the peak valuesof the component sinusoidal functions are displaced relative to eachother. The effect of this can best be seen in the unrolled view in FIGS.6 and 7 wherein the quadrupole amplitude A₂ is selected to be half thedipole amplitude A₁. The phase shift Δφ is zero in FIG. 6 and is 90degrees in FIG. 7. That is, in FIG. 7, φ_(j)−φ_(i)=90 degrees. Theconductor spacing, h, for each case is set to the required minimumvalue.

A feature of the invention is that the maximum value of the slope angleα, referred to as α_(max), is a function of the relative phase shiftbetween components of different orders, n, and this can lead to adecrease of the maximum slope angle α_(max) thereby reducing the minimumachievable conductor spacing h and increasing overall conductor densityalong the axis. This enhances the magnetic field density. For the givenexample with A₂ equal to one half A₁, the minimum achievable conductorspacing can be reduced by about ten percent. Increasing the conductordensity increases the magnetic transfer function, thereby increasing thefield magnitude per unit of current. More generally, useful improvementsin the transfer function can be realized in combined function assemblieswhere, for individual coil rows, X(θ) includes at least the followingterms:[h/(2*π)]θ+A _(i) sin(θ)+A _(j) sin(jθ+Δφ)+ . . .In example embodiments, A_(i) is at least ten percent of A_(j).

In a series of embodiments according to the invention, with windingsconfigured in accord with Equations 3A, 3B and 3C to produce relativelyuniform dipole fields, such winding patterns can be adapted forapplications in which a charged particle beam path includes a curvedportion in order to conform with a desired trajectory. Conventionally,one might couple together several sections of saddle coil windings withthe sections coupled end to end at modest angles to create a curvedpath. Similar arrangements can be effected with multiple double helixwinding segments, each formed along a straight axis.

The straight double helix segments can be configured to generate auniform dipole field, with substantially no higher order fieldcomponents, in order to displace the beam of charged particles along adirection transverse to the straight axis. With adjoining segmentscoupled at small angles to create such a bend, each angle corresponds toan arc length or a linear distance across the aperture through which thebeam is displaced. In effect, the beam is made to follow a curvedtrajectory while traveling through a series of straight apertures. Withthis design approach the minimum radius of curvature about a bend isdependent on the beam aperture size. However, by using a Double Helixwinding for this application it is possible to provide a highly uniformdipole moment (n=1) and to also incorporate a quadrupole moment to helpfocus the beam.

Instead of coupling segments of magnets having straight apertures, i.e.,segments only having straight apertures, magnets with curved aperturesmay also be manufactured with helical coils. Although straight magnetsbased on helical coil designs according to Equations 3A, 3B and 3C canhave essentially pure fields, e.g., very uniform transverse fields of adesired order, the formation of a helical coil design about a curvedaxis will normally generate, in addition to a primary field of desiredorder, one or more higher order components which cause a departure fromthe ideal field properties in a curved beam trajectory. According toseveral embodiments, by introducing further modulating components intohelical conductor geometries, e.g., into individual coil loops, theundesirable components, which would otherwise degrade the field qualityof a magnet, can be nearly or completely canceled. That is, multipolefield components can be introduced to counter each of the undesirablehigher order components, these each being of practically equal magnitudeand opposite direction to an undesirable component. With numericalcomputation and optimization techniques, one or more multipolecomponents of desired proportion can be introduced with precision. Thesearrangements can be implemented by using a combined function feature ofdouble-helix coils. Implementations of these concepts provide forsequential variations in field properties along an axis of variablecountour. For the example of an axis having a constant radius ofcurvature, proportions of the main field components may vary as afunction of position along the axis so that along a first of threeadjoining segments, a dipole field may predominate while along anadjacent second segment a quadrupole field may be the primary component,and along a third segment adjoining the second segment another dipolefield may be the main field component. Generally, the relativeproportions of field components may be varied to provide desired effectssuch as combinations of beam focusing and bending. Also, generally, useof the term “main field component” in the singular or plural refers to amultipole field component which is a dominant field component along oneor more planes transverse to a central axis along which a coil row orentire magnetic assembly is formed.

In the example shown in FIG. 8, a double helix dipole magnet 10 has aradius of curvature of 225 mm along a central axis and an apertureradius of 25 mm. Consistent with double helix design principles, twocoil rows 12A and 12B of the magnet 10 are illustrated as havingopposite tilt angles with respect to transverse planes along the centralaxis. The magnet 10 comprises multiple pairs of similar coil rows 10(not shown) formed over one another in a manner analogous to what hasbeen described in Ser. No. 12/061,813 for an assembly formed along astraight axis. The geometry of the magnet 10 can be used to impart a 90°aperture bend but other radii of curvature, aperture lengths andaperture radii are readily achievable in accord with the now-describedmethodology. First, it is noted that designing such a magnet to suchgeometric specifications and desired field properties could involvecalculation of conductor paths about a curved axis based directly onfield specifications along a curved axis of symmetry. For example, basedon the equations 3A, 3B and 3C,X(θ)=[h/(2*π)]θ+ΣA _(n) sin(nθ+φ _(n))  3AY(θ)=R cos(θ)  3BZ(θ)=R sin(θ)  3Cit is possible to perform appropriate transformations into a curvilinearsystem with which the transformed analytics are used to generate a setof points which describe a helical path about a curved axis of symmetry.With the path divisible into a series of conductor segments about acurved aperture, numerical techniques can be used to calculate themagnetic field about the aperture. That is, with the conductor pathdivisible into the segments, the field may be calculated at all pointsof interest by adding the contribution of each segment to each point ofinterest. Such an approach allows for the determination of undesirablehigher order field components.

With the foregoing transformation of the analytics or, as described,when coil patterns according to a straight axis double helixconfiguration are mapped into curvilinear coil patterns, to create acurved magnet geometry, higher order multipole field components areintroduced. In principle, these effects can be compensated usingmathematics in a curvilinear system by introducing appropriate changesin the modulation of the curvilinear coil patterns. Undesired multipolefield components could be directly removed by adding offsettingmultipole components directly into the analytical equations whichdescribe the curvilinear coil patterns. In practice, the volume andcomplexity of numerical calculations required to perform thesecorrections, whether in rectangular or a curvilinear coordinate system,can make it a lengthy iterative process to directly determine thenecessary compensating components that offset undesired higher ordermultipole field components.

Instead of adding correction factors to the analytics describing acurvilinear coil pattern, the undesired multipole components, e.g.,components introduced by bending from the straight geometry, usefuladjustments for the curvilinear coil patterns can be created in theuntransformed coil configuration as defined along a straight axis.Another feature of the invention is based on recognition thatsubstantial orthogonality is preserved among multipole components inboth the straight and curvilinear systems. Numerous embodiments of coildesigns may be developed based on point-to-point transformations betweenpatterns formed on straight and curved axes. For example, adjustmentsneeded to cancel specific multipole field components in the design for acurvilinear coil pattern can be based on corrections made to the modelof a corresponding straight axis coil configuration from which thecurvilinear coil pattern is derived.

Thus, for the first time, the design for a straight geometry coilconfiguration can be applied to design coil patterns having a curvedgeometry with the reduction or elimination of all undesired multipolefield components. Within a desired degree of tolerance, a helical coilmagnet with a curvilinear aperture can be constructed along a curvedaperture with substantially the same multipole field quality as a magnetconstructed along a straight axis. Additional multipole componentssuitable for the application may also be introduced to the analytics forthe straight geometry so that, after transformation into the design forthe curved geometry, the magnet exhibits substantially the samemultipole field quality as a magnet constructed along a straight axis.

For illustrated embodiments a method of achieving the desired fieldquality along a curved axis may include a repetitive process ofperforming transformations between the straight and curvilinearanalytics, with modifications to the straight geometry analytics untildesired field properties are attained for a magnet constructed along thecurved aperture. As used herein, field quality refers to measures of therelative magnitudes or uniformities among multipole field components ina magnet. Examples follow. A detailed process is now described fordesign and construction of magnets, including the magnet 10 of FIG. 8,with optimized multipole content.

The above-described parametric representation, in terms of X(θ), Y(θ),Z(θ) describes the space curve for a single layer, i.e., one helicalshaped coil row comprising a continuous series of open conductor loops.In the following descriptions, although reference is made to spacecurves for individual layers, it is to be understood that actual designsand the construction of magnets accordingly involve implementing thedescribed processes to create multiple coil rows formed about oneanother. The fields from each row are additive such that a model of theconductor path can be constructed for each coil row and the generatedfield can be determined by adding contributions generated by each.Reference to generated fields in the several examples may be understoodto mean the aggregate fields generated by multiple coil rows or thefield components generated by a single coil row. For purposes of moreclearly presenting features or advantages of the invention, the fieldcalculations presented in Tables 1, 2 and 3 are representative ofcalculated field strengths made along points on the axes which are atleast three aperture diameters from the coil ends. This substantiallyavoids inclusion of contributions stemming from uncompensated higherorder terms characteristically found around the coil ends of manymagnets.

The magnetic field in a long straight section of a cylindrical-shapedhelical configuration, generating a transverse field, can be consideredas two dimensional and can be described in a cylindrical coordinatesystem in accord with the following harmonic expansion:

${B_{\theta}\left( {r,\theta} \right)} = {B_{ref}{\sum\limits_{n = 1}^{\infty}{\left( \frac{R}{R_{O}} \right)^{n - 1} \cdot \left( {{b_{n}{\cos\left( {n\;\theta} \right)}} + {a_{n} \cdot {\sin\left( {n \cdot \theta} \right)}}} \right)}}}$${B_{\theta}\left( {r,\theta} \right)} = {B_{ref}{\sum\limits_{n = 1}^{\infty}{\left( \frac{R}{R_{O}} \right)^{n - 1} \cdot \left( {{b_{n}{\sin\left( {n\;\theta} \right)}} - {a_{n} \cdot {\cos\left( {n \cdot \theta} \right)}}} \right)}}}$wherein R_(o) is the reference radius measured from the axis and B_(ref)is the magnitude of the main field at this radius. The coefficientsb_(n) and a_(n) are dimensionless normal and skew multipole componentsthat determine the angular orientation of the different componentsrelative to each other.

Although the harmonic expansion describes a two-dimensional field alongan infinitely long axis, it is also convenient to characterize the endfields of a magnet of limited length with the same harmonic expansion.In this case, along axial positions that are less than three aperturediameters from the coil ends, the two dimensional multipole fieldcomponents in planes transverse to the axis will vary as a function ofthe axial position, while at further distances from the coil ends thefield components are relatively constant. This assumes that the coilextends along the axis at least a distance of six aperture diameters.Also, within three aperture diameters of the coil ends, there is anon-zero axial field component B_(x), which can be determinedindependently.

When numerically computing multipole fields in the magnet it may beassumed that the conductor path can be represented by an infinitely thinfilament located at the center of the physical conductor, which in manyinstances may have a circular cross section. Square orrectangular-shaped conductors can also be modeled by placing the thinfilaments in the conductor cross section in such a way that theyapproximate the current distribution within the conductor. Athree-dimensional space curve of a conductor may be described as apolygon of small straight filament sections. The end points of eachpolygon segment will coincide with the actual space curve, and for asufficiently large number of elements, the polygon describes the spacecurve with a high degree of precision.

For an infinitely thin and short filament of conductor segment, themagnetic field at any point in space not coinciding with the conductorsegment is given by the Biot-Savart Law as follows:

$\overset{\rightarrow}{dB} = \frac{{I \cdot \overset{\rightarrow}{d\; 1}} \times \overset{\rightarrow}{P}}{P^{2}}$where dB is the field vector, I is the transport current flowing throughthe conductor, dl is the tangent vector of the filament segment and P isthe vector from the segment to a point at which the field is to bedetermined. By summing the field contributions from all segments alongthe space curve describing the conductor path, an approximation of themagnetic field is obtained. The accuracy of the approximation increasesas the conductor is divided into a greater number of segments for whichindividual computations dB are performed.

The multipole content can then be determined in the following way. For acoil row formed along a straight axis, in a plane transverse to theaxis, the field components B_(r) and B_(θ) are calculated for n points Pequally spaced along the azimuth of a reference circle formed in theplane and centered about the axis. A Fourier analysis of these fieldvalues with appropriate normalization yields the multipole fields inunits of Tesla or Gauss. The axial field component may be neglected forthe purposes of assessing or modifying the transverse field componentsin a double helix coil configuration. The same multipole field analysiscan be performed for a series of circles (each in a different planetransverse to the axis and intersecting different positions) along astraight or curved magnet axis.

For the single layer space curve corresponding to one coil row in a puredipole double helix magnet formed along a straight axis,X(θ)=[h/(2π)]θ+A ₁(sin θ).Such a single layer, formed along a straight axis, contains an axialfield component. Recalling, also, thatY(θ)=R cos(θ)Z(θ)=R sin(θ)the axial field component can be canceled by adding a concentric coilrow having a slightly different radius. With the first and second coilrows having opposite tilt angles and current directions, the two coilrows form a pair which provides an equal but opposite axial current flowthat cancels out. Exemplary pairs of coil patterns generated accordingto this equation can be transformed into coil patterns along a curvedaxis of symmetry. See FIG. 9 in which a two-dimensional Cartesiancoordinate system is shown to include an X axis and a Y axis whichintersect with one another at (x,y)=(0,0). FIG. 9A illustrates thepattern of a straight helical coil row 20 in a pair of rows formed alonga straight axis 24 coincident with the X axis and also illustrates thestraight row transformed into a curved double helix coil row 20′ formedalong a curved axis 24′. As shown in FIG. 9B the coil row 20 has theshape of a regular cylinder with an inner radius y=y₁ corresponding tothe coil aperture. In this simple example the coil row 20 is transformedinto a geometry having a circular radius of curvature, wherein R′corresponds to the bending radius of curvature along a center linecoincident with the Y axis. In the transformation the straight X axis 24becomes the curved axis 24′ which is an axis of symmetry for the coilrow 20′. In the plane positioned on the Y axis and orthogonal to the Xaxis, cross sections of the row 20 and the row 20′ completely coincide.The curved axis 20′ is symmetric about the Y axis (x=0), having an innerradius of curvature, R₁′=R′−y₁, and an outer radius of curvature,R₂′=R′+y₁.Next, referring also to FIG. 9B, for an arbitrary point P₁ (x₁,y₁) whichmaps into a point P₁′(x₁′,y₁′), the distance, x₁, of the point P₁ alongthe X axis (relative to x=0) is the arc length along the radius R₁between the origin (0,0) and the point P₁′(x₁′,y₁′).

For the transformation of a given point (x₁, y₁) the bending radiusdepends on the y-coordinate.R ₁ ′=R′−y ₁Arc₁ =x ₁ =R ₁′*ΔΨ₁ΔΨ₁ =x ₁ /R ₁ ′=x ₁/(R′−y ₁)The resulting transformation is:x ₁ ′=R ₁′ sin(ΔΨ₁)y ₁ ′=y ₁ +R ₁′(1−cos(ΔΨ₁))z ₁ ′=z ₁

Through this bending transformation changes in the multipole fieldcontent appear attributable to a conductor spacing which is smaller onthe inside of the arc than the conductor spacing on the outside of thearc having radius (R′+y₁). Increased wiring density, i.e., distancebetween adjacent portions of turns, along the inside of the arc, havingradius (R′−y₁), creates a higher field in that vicinity, relative todecreased wiring density, along the outside of the arc, which results ina relatively lower field. Such an asymmetric wiring pattern about thecoil axis is believed to result in generation of higher-order multipolecomponents. The main effect of transforming a pattern from a straightaxis to a curved axis is a field gradient which primarily creates aquadrupole component. When a pure dipole moment is desired for thetransverse field along the curved axis 24′, the higher-order termsresulting from the bend can be offset by introducing additional momentsthat cancel the components introduced by the bending transformation.This compensation can effectively be designed by introducinghigher-order multipole components into the pattern for the straight coil20 and then transforming the modified pattern of coil 20 into a revisedpattern for the coil 20′. By choosing the signs and amplitudes of thesecomponents appropriately, the curved coil row 20′ can generate asubstantially pure dipole field. The correction to remove the undesiredfield gradients can be determined as an iterative process until theundesired components are reduced to a factor which is multiple orders ofmagnitude smaller than the dipole moment.

FIG. 10 illustrates an exemplary procedure which can be applied todetermine the higher-order terms which are to be introduced into thestraight winding in order to offset the higher-order terms resultingfrom the transformation which generates the coil 20′. Initially thespace curve conductor patterns are generated for each coil row 20 formedalong the straight axis 24. For a pure dipole field the initial valuesof each ⊖_(n) would be zero. Depending on the multipole moments desiredin the curved magnet geometry, the space curve can be made a function ofone or multiple ones of the parameters ⊖_(n). Also, during the iterativeprocess of FIG. 10, various ones of the terms ⊖_(n) may be identified asparameters which are to be adjusted in order to remove unwantedmultipole components.

A bending transformation is applied to the coil row space curves togenerate a curved magnet geometry defining each coil row 20′. This isfollowed by calculation of the multipole content of each coil row 20′,individually or collectively. The multipole content may be determinedwith the Fourier analysis described above. For each of the componentsfound in the analysis there is an associated value ∈_(n). Based on thisanalysis and specified design criteria, certain multipole fields areidentified as unwanted and an objective function is generated to developmodulations in the wiring pattern in order to generate field componentswhich offset the undesired fields. That is, a term ∈_(n) exists for eachunwanted field component, and an objective function is structured tofind values of each ∈_(n) to offset unwanted field components. Whenincorporated into Equations 3, the values of each ∈_(n) generated by theobjective function will contribute to generation of a field componentthat offsets one of the unwanted field components. An objective functionmay be structured to provide a net minimum value field magnitude for agiven multipole order when the strength of an unwanted field falls belowa predetermined value. The parameters ∈_(n) are modified for relevantvalues of n, based on a search which seeks to minimize each unwantedfield component. Once this is done the process may be repeated based ona transformation of a new space curve (including the values of ∈_(n)generated by the objective function) from a straight coil row 20 into acoil row 20′ along the curved axis 24′.

More generally, the process can be summarized as first defining thespace curve for a coil row along a straight axis according to a set ofparameters which meet specifications and then doing a transformation ofthe curve into a desired curvilinear geometry. A determination is madeof deviation from the specifications. The deviation is minimized byvarying parameters that affect modulation of the space curve, e.g.,according to the process of FIG. 10.

In another example illustrated in FIG. 11, a double-helix coil magnet 16is formed along a straight axis consisting of two coil rows 18A and 18Bin accord with equations 3A, 3B and 3C and n=1, each row having anexemplary 100 turns and a coil aperture radius of 25 mm. As used hereinthe term coil aperture radius corresponds to the distance R in theequations 3B and 3C for the inner coil row 18A. The distance R, used inthe equations 3B and 3C for the coil row 18B, has a value of 27 mm.Generally, in the described embodiments, the distance R corresponds tothe distance between the axis of symmetry along the coil and the centeraxis of the conductor, e.g., a round wire, used to generate the coilpattern. Also, it is assumed that the usable aperture is actuallysomewhat smaller than the value of R for row 18A to account for wirethickness and coil support and possible other intervening materials.Magnets based on bending transformations relative to the magnet 16 mayhave relative spacings between coils rows, e.g., rows 12A and 12B, whichare consistent with the relative spacings described for the straightcoil rows 18A and 18B. That is, even though the rows 12A and 12B areformed about a curved axis, the differences in distance of each of theserows relative to the central axis is consistent with the exampledistances of 25 mm and 27 mm. However, other embodiments may be based inpart on variations in such spacings.

Tables 1a through 1d list for the magnet 16 the corresponding calculatedmultipole content along perimeters of circles transverse to the axis toillustrate dependence of multipole fields as a function of radialdistance from the axis. The multipole content is calculated along aplane passing through the center point of the magnet axis which is alsothree aperture diameters from each coil end. The circles are of varyingradius relative to the 25 mm aperture radius, i.e., 5 mm, 10 mm, 15 mmand 20 mm with field calculations based on a coil current of 236 A whichgenerates a dipole field of 1000 Gauss. Tables 1 also illustrate thatthe relative magnitudes of the higher-order multipole components (inparticular, compared to the dipole component) increase with increasingdistance from the axis. However, it can be seen that even at a 25 mmreference radius, which is 80 percent of the coil aperture radius, thehigher-order multipole fields are still about one part per million ofthe main dipole field.

TABLE 1a Multipole content for straight coil 16 at reference radius R =5 mm. The dipole field of 1000 Gauss is based on a 236 A coil current.MP Order An Bn Cn 1 1.78E−02 1.00E+03 1.00E+03 2 −1.69E−04 2.42E−042.95E−04 3 −1.25E−06 −5.03E−05 5.03E−05 4 −8.56E−09 1.45E−07 1.45E−07 5−3.60E−11 1.59E−07 1.59E−07 6 −3.22E−14 5.77E−10 5.77E−10 7 2.36E−12−4.68E−08 4.68E−08 8 −4.31E−13 −3.23E−10 3.23E−10 9 −2.07E−12 −2.81E−092.81E−09 10 9.49E−13 −6.21E−12 6.29E−12

TABLE 1b Multipole content for straight coil 16 at reference radius R =10 mm. The dipole field of 1000 Gauss is based on a 236 A coil current.MP Order An Bn Cn 1 1.78E−02 1.00E+03 1.00E+03 2 −3.35E−04 4.79E−045.85E−04 3 −4.95E−06 −1.97E−04 1.97E−04 4 −6.76E−08 1.16E−06 1.16E−06 5−5.58E−10 2.54E−06 2.54E−06 6 −1.18E−11 1.85E−08 1.85E−08 7 6.43E−11−3.00E−06 3.00E−06 8 −1.34E−12 −4.14E−08 4.14E−08 9 −9.58E−12 −7.20E−077.20E−07 10 1.79E−12 −3.02E−09 3.02E−09

TABLE 1c Multipole content for straight coil 16 at reference radius R =15 mm. The dipole field of 1000 Gauss is based on a 236 A coil current.MP Order An Bn Cn 1 1.80E−02 1.00E+03 1.00E+03 2 −4.97E−04 7.08E−048.65E−04 3 −1.09E−05 −4.25E−04 4.25E−04 4 −2.23E−07 3.92E−06 3.93E−06 5−2.70E−09 1.29E−05 1.29E−05 6 −9.15E−11 1.41E−07 1.41E−07 7 7.41E−10−3.41E−05 3.41E−05 8 6.14E−12 −7.18E−07 7.18E−07 9 −2.72E−10 −1.85E−051.85E−05 10 −2.58E−12 −9.52E−08 9.52E−08

TABLE 1d Multipole content for straight magnet 16 at reference radius R= 20 mm. The dipole field of 1000 Gauss is based on a 236 A coilcurrent. MP Order An Bn Cn 1 1.82E−02 1.00E+03 1.00E+03 2 −6.51E−04−2.51E−04 6.98E−04 3 −2.09E−05 −1.22E−03 1.22E−03 4 −1.13E−06 8.37E−048.37E−04 5 3.78E−07 −7.92E−04 7.92E−04 6 2.20E−07 4.43E−04 4.43E−04 71.34E−06 3.06E−04 3.06E−04 8 1.01E−06 −1.10E−03 1.10E−03 9 −2.52E−06−4.91E−04 4.91E−04 10 −1.15E−06 1.52E−03 1.52E−03

According to an embodiment of the invention, a bending transformation isapplied to the straight coil rows, converting them into a curvedgeometry which covers an arc section of 90 degrees. In this example, theresulting winding pattern conforms with the shape of the magnet 10 ofFIG. 8. The curved coil rows 12A and 12B are each based on atransformation of one of the straight rows 18A or 18B. For the dipolecoils 12A and 12B, prior to inclusion of any additional moments tocancel or offset higher order terms resulting from the bendingtransformation, the calculated multipole field strengths are shown inTable 2. As can be seen a strong quadrupole component B₂ has developed,which is about two percent of the main field B₁. Additionally, smallerhigher-order terms have been introduced.

TABLE 2 Multipole content for the magnet 10 formed along a curved axisat a reference radius of 20 mm (80% of coil aperture). The skew dipolecomponent A_(n) (n = 1) has been adjusted to 1 × 10⁻⁷ by introducing asmall phase angle of 2 × 10⁻⁵. The dipole field of 1000 Gauss is basedon a 236 A coil current. MP Order An Bn Cn 1 2.61E−07 1.00E+03 1.00E+032 −2.49E−03 −2.03E+01 2.03E+01 3 −2.04E−04 −1.07E+00 1.07E+00 4−1.58E−05 −5.18E−02 5.18E−02 5 −9.45E−07 −3.96E−03 3.96E−03 6 −3.56E−073.81E−04 3.81E−04 7 4.28E−07 8.15E−04 8.15E−04 8 9.01E−07 −1.05E−031.05E−03 9 −1.44E−06 −2.88E−04 2.88E−04 10 −8.00E−07 5.42E−04 5.42E−04

Noting that the X(θ) equation for each of the coils 18A and 18B producesa pattern which generates a dipole field based on presence of a termA₁(sin θ), the process for reducing all of the higher order fieldcomponents introduced by the bending transformation begins withintroducing to the X(θ) equation additional components:X(θ)=[h/(2π)]θ+A ₁(sin θ+∈₂ sin(2θ+Δφ₂)+∈₃ sin(3 θ+Δφ₃)+∈₄sin(4θ+Δφ₄)),  (6)wherein each additional component is of an order that is the same as anorder of a component which is to be offset. As can be seen from equation6, describing X(θ), the modulation along a straight axis can consist ofa sum of multiple sinusoidal components, with each component having aweighting factor ∈_(n). Once a determination is made as to which of thecomponents of the multipole orders, introduced by the transformation tothe curved geometry, are to be reduced or eliminated, a correctioncomponent for each such multipole component can be incorporated into theequation for X(θ), i.e., by assigning an appropriate weighting. Theoffsetting of these undesired component magnitudes is had by giving theadded correction components signs opposite those of the undesiredcomponents.

In this example the offsetting terms introduced are limited to thequadrupole component ∈₂ sin(2θ+Δφ₂), the sextupole component ∈₃sin(3θ+Δφ₃) and the octupole component ∈₄ sin(4θ+Δφ₄). Using the wellknown Simplex algorithm the six corresponding parameters ∈_(n) andΔφ_(n) (for n=2, 3 or 4) are optimized in such a way that the undesiredmultipole fields resulting in the bent coil 10 from the transformationbecome offset, i.e., the net magnitude for each higher order componentis reduced. As can be seen in Table 3 the components corresponding tomultipole orders 2, 3 and 4 are thereby suppressed. The optimizedparametric values of ∈_(n) and Δφ_(n) are listed in Table 4. Allmultipole field components that have been optimized are reduced tolevels more than ten orders of magnitude smaller than the magnitude ofthe main field component. Table 3 confirms that higher multipole orderscan be reduced by 13 to 15 orders of magnitude relative to the mainfield component. Understanding that some applications do not requiresuch high field quality, embodiments include suppression of multipolefield components by 3 orders of magnitude or higher, e.g., 4, 6, 8, 10or more orders. The design process may comprise multiple cycles ofiterative optimizations, where in each cycle a new set of fieldcalculations are performed on a curved pattern and an objective functionis optimized by modifying the straight geometry until the magnitudes ofthe undesired higher order components fall below a predetermined levelof acceptability.

TABLE 3 Net multipole content for the magnet 10 formed along a curvedaxis at a reference radius of 20 mm (80% of coil aperture) afterintroduction of offsetting components to suppress quadrupole, sextupoleand octupole fields. The current remained at 236 A. MP Order An Bn Cn 1−1.05E−06 1.00E+03 1.00E+03 2 1.98E−10 4.61E−12 1.98E−10 3 1.65E−10−1.38E−10 2.16E−10 4 −3.59E−11 −1.29E−10 1.34E−10 5 −1.32E−06 −4.85E−034.85E−03 6 −1.19E−06 8.94E−05 8.94E−05 7 8.66E−07 −4.58E−05 4.59E−05 86.87E−07 3.22E−04 3.22E−04 9 −1.41E−06 6.31E−04 6.31E−04 10 −4.07E−07−1.89E−03 1.89E−03

TABLE 4 Optimized Parameters resulting in the filed components listed inTable 3. ε_(n) Δφ_(n) ε₂ sin (2θ + ΔΦ₂) 1.358833E−02 1.227936E−04 ε₃ sin(3θ + ΔΦ₃) 7.656867E−04 1.786302E−04 ε₄ sin (4θ + ΔΦ₄) 4.299757E−052.413683E−04

After performing the optimization to determine values for theparameters, the X(θ) equation for each of the wiring patterns in thestraight double helix coil 16 becomes as follows:

X(θ) = [h/(2π)]θ + A 1 * (sin (θ) + 1.358833 e-2 × sin (2 * θ + 1.227936 e-4) + 7.656867e-4 × sin (3 * θ + 1.786302 e-4) + 4.299757 e-5 × sin (4 * θ + 2.413683 e-4)).

Changes in the decapole field component, after optimization ofquadrupole, sextupole and octupole fields (compare values in Tables 2and 3), indicate that in performing the adjustment procedure multipolecomponents of different orders do not behave completely independentlyfrom one another, i.e., they do not exhibit completely orthogonalbehavior during the suppression process. Upon adding modulationcomponents corresponding to lower orders, the field of the decapoleorder (B5) changed to −4.85E-03 from a value of −3.96E-03. However, ifthe optimization procedure included these orders as well, they wouldalso be substantially reduced.

The procedure described here for the adjustment or suppression ofcertain multipole components has various applications. Magnet coils areoften equipped with surrounding iron yokes, which can enhance the fieldgenerated by the coil or are needed to limit the fringe magnetic field.Iron saturates at high magnetic fields, and the field enhancement due tothe iron yoke saturates. This leads to higher-order multipole componentsat high levels of magnet excitation. If the field uniformity of such amagnet is most important at high field levels, the describedoptimization procedure can be used to offset undesired higher-orderterms. As for curved geometries, higher-order offsetting terms can beintroduced into the coil geometry in accord with the equationX(θ)=[h/(2π)]θ+A ₁(sin θ+∈₃ sin(3θ+Δφ₃)+∈₅ sin(5θ+∈φ₅+ . . . )),  (1)with the parameters optimized to compensate field components caused byiron saturation at a certain coil excitation. More generally, both ironsaturation effects and superimposed error fields (e.g., caused byinterfering magnetic systems of materials) can be offset in a coilassembly. These types of corrections can also be implemented for otheraperture geometries described herein.

The optimization approach can additionally be used to compensate forfield errors that are introduced by unavoidable manufacturingtolerances. The performed optimization shows that systematic fielderrors, i.e., those intrinsic to the design of the conductor pattern canbe controlled with very high accuracy. However, the suppression ofhigher order multipole fields that can be achieved in a magnet alsodepends on the manufacturing accuracy, i.e., with the degree ofprecision of conductor placement. Since conductors can only be placedwith certain accuracy in coil manufacturing, unwanted higher-order fieldcomponents will result. In the above example, for a coil with a 25 mmaperture, the modulation amplitude generating the main dipole field isalso 25 mm. The optimized parameter ∈₂ correcting the unwantedquadrupole component (see Table 4) is 1.36×10⁻². Multiplying this factorwith the main amplitude amounts to about 0.35 mm. The smallest amplitudemodulation that can be realized in machining of support grooves ismainly determined by the resolution of the angular encoders and motorsof the machining centers of a CNC machine and, to a lesser extent, byabsolute accuracy of the machine. It is therefore possible to implementmodulations of the main amplitude down to a level of 0.001 mm.

Embodiments of the invention are based in part on recognition thatindividual field components of differing orders in a helical coilconfiguration may not be completely orthogonal to one another. Notingagain that the decapole order B₅ shown in Table 3 became larger when theoptimization was only preformed for three other orders, it is recognizedthat individual multipole orders cannot always be adjusted withoutaffecting one or more other multipole components of different order. Itis with simultaneous optimization of all relevant parameters, e.g., formultiple relevant multipole orders, that satisfactory reduction of allundesired field components can be assured.

Thus, for the first time, the design for a straight geometry coilconfiguration can be applied to design coil patterns having a curvedgeometry with the reduction or elimination of all undesired multipolefield components. Within a desired degree of tolerance, a helical coilmagnet with a curvilinear aperture can be constructed along a curvedaperture with substantially the same multipole field quality as a magnetconstructed along a straight axis. Additional multipole componentssuitable for the application may also be introduced to the analytics forthe straight geometry so that, after transformation into the design forthe curved geometry, the magnet exhibits substantially the samemultipole field quality as a magnet constructed along a straight axis.For illustrated embodiments the method of achieving the desired qualitymay include an iterative process of performing transformations betweenthe straight and curvilinear patterns with modifications to theparameters describing the straight geometry until calculations indicatethat desired field properties are attained for a magnet constructedalong a curved aperture. Field quality is directly based on the relativemagnitudes of multipole field components.

In summary, for the example of FIG. 8, design of the double helix coil10 having a 90° bend begins with a first transformation of the analyticsfrom the straight geometry to the geometric specification for a magnethaving the curvilinear coil aperture. This is followed by fieldcalculations which indicate the extent to which undesired higher orderfield components are present due to the direct transformation of thecoil from a straight axis to a curved axis. In the example of FIG. 8,modeling of the field generated along the curved axis confirms that aquadrupole moment and, to a lesser extent, higher order multipolecomponents have been introduced. Analysis of the transverse fieldindicates presence of a quadrupole component that is approximately twopercent that of the dipole field along the perimeter of a circle havinga reference radius of 20 mm, i.e., 80 percent of the 25 mm apertureradius. See again Table 2. The field calculations are representative ofvalues at positions along the axis which are at least three aperturediameters from the coil ends.

Next, a determination is made of an adjustment to the analytics togenerate an appropriate quadrupole component that offsets the quadrupolemagnitude by about 20 Gauss relative to the dipole field. This is basedon the relationship shown in equation 6, wherein modulation factors∈_(n) are expressed as normalized magnitudes relative to A₁.

The foregoing demonstrates that the combined function capability of ahelical coil design can suppress or tune out an undesired quadrupolemoment when creating a curved geometry without degrading the strength orquality of the primary, e.g., dipole, field. The adjustment process canalso be applied to a complete magnet with yoke to tune out saturationinduced multipoles as required.

Generally, a helical coil formed about a curved axis can be designed toproduce a substantially pure field, e.g., a dipole field, of a givenorder, or to produce combinations of superimposed multipole fields. Bydesign, one or more field components can be made many orders ofmagnitude larger than other field components. Pure dipole fields canalso be produced when an iron yoke surrounds the coils, especially athigher field levels. The coil winding pattern on a helical path can bemodified to practically eliminate saturation induced multipoles, e.g.,in a dipole magnet.

The compensation of higher order fields can be implemented in amanufacturing process. Placement of the conductor in precisely machinedgrooves, as described in co-pending application Ser. No. 12/061,813assures that desired introduction of modulation components in each loopis of sufficient accuracy to effect the field cancellations. Further,the placement of conductor in machined grooves is advantageous for themanufacturing of magnets having helical shaped coils formed about curvedaxes because slippage, or movement, of the conductor under the effectsof strong Lorentz forces and temperature cycling, is mitigated orcompletely avoided.

Stabilization of conductors forming a curved coil geometry is of utmostimportance for normal conducting and superconducting windingconfigurations. In the past conventional winding technology has not beenapplied to curved magnet geometries, e.g., wherein part or all of theaperture is formed along a curved axis of symmetry. Nor is it suitableto wrap a winding around a curved support structure like in a solenoidor window-frame type configuration because conductors so deployed in abent geometry can easily shift or slip over the surface of the supportstructure. Technology to hold the conductor in position is needed toachieve precise and stable conductor placement. In high fieldsuperconducting magnets this requirement is more stringent than forresistive magnets since larger Lorentz forces are present to act on theconductor. Frictional movement due to slippage of the conductor caninitiate quenching of a superconducting coil, in which a rapidtransition from the superconducting to the normal conducting phase takesplace. Machining of support grooves or channels into the surface of thesupport structure is a reliable way of achieving precise conductorplacement with high mechanical robustness. See, again, Ser. No.12/061,813.

FIGS. 13A-13G illustrate fabrication features for construction of thecoil 10 according to embodiments of the invention. The designincorporates multiple layers providing pairs of tilted double helixconductors configured about the curved axis 24′. FIG. 13F provides aview in cross section of the coil 10 along an arbitrary plane cuttingtransversely through the axis 24′ and illustrating the aperture 102formed within a core 104. The coil 10 includes multiple coil rows 12,including rows 12A and 12B shown in FIG. 8, other specific ones of whichare also referred to herein with other reference numbers. Each row 12comprises a helical conductor formed in an insulative layer. In planestransverse to the axis 24′ each of the coil rows 12 is concentricallyplaced about the axis 24′ and with respect to the other rows 12.Portions of the conductor in different ones of the rows 12 areelectrically isolated from one another by one or more layers of theinsulator as illustrated in the figures. The various layers of insulatorwhen referenced generally or collectively are referred to as layers 108.Other reference numbers are used when referring to specific layers ofinsulator in the coil 10. The insulator may be a relatively rigidnon-conducting composite material in which channels can be machined forstable positioning of the conductor in each coil row. However, theinvention is not at all limited to such designs or to the arrangement oftilted helical patterns shown for the coil 10.

One exemplary fabrication sequence, suitable for manufacturing numerousembodiments of coils begins with formation and curing of a layer 108 ofcomposite material about the core 104. The core may be a removablemandrel as shown in the figures or may be a permanent structure, such asa stainless steel cylinder which provides a beam tube during use of themagnet in a particle accelerator application or for creating a vacuum inthe aperture. The mandrel may be dissolvable or chemically removable. Inother embodiments, the core 104 may be a composite material formed, forexample, of fiberglass resin and suitable for formation of one or morechannels therein to define a core coil row, e.g., row 12A. When multiplechannels are formed in the same row they may be interlaced with onechannel providing an auxiliary function such as cooling of the conductor(not shown). The mandrel or the core may be insulative or conductivebodies. The illustrated core 104 is a removable shaft suitable formounting on a Computer Numerical Control (CNC) machine in an automatedprocess. A core may be formed of ceramic, composite material or othermoldable or machinable material. Although illustrated embodimentssuggest cores which are circular in cross section and of uniformaperture diameter, other geometries are contemplated.

As shown in FIG. 13A, a layer 108 of insulator comprising a compositematerial is formed on the core 104. Such a so-called lay-up may, forexample, be a reinforced plastic comprising fibers, e.g., fiberglass,carbon or aramid, and a polymer, such as an epoxy or a thermosettingplastic. The layer 108 may be applied as a series of sublayers eachcomprising a thin, chopped strand or woven fiber mat through which aresin material permeates, or as a matrix of fiber particles and polymer.Thickness of the layer 108 is chosen based on numerous considerationsincluding the thickness or diameter of the conductor to be placed on thelayer, the desired depth to which the conductor is to be placed in thechannel, and the minimum thickness of insulation between conductormaterial positioned in adjacent ones of concentric rows. The compositelayer 108 is cured in a conventional manner and then machined to desiredtolerances. A substantial portion of the cured and machined layer 108 isin the shape of a curved tube also referred to herein as a curved orbent cylindrical body 110. See again FIG. 13A, which illustrates thatthe body 110 has a major outer surface 118 which when viewed along aplane transverse to the axis 24′ is circular in cross section. However,asymmetric geometries of the resulting coil may be fabricated in asimilar manner. Generally, the layer 108 may be any tubular shape,having in some embodiments a central axis of symmetry or multiplethicknesses or variable shapes along an exterior surface. The aperture102 within the layer 108 may also be in the shape of a circle (i.e.,when viewed in a cross section taken along a plane transverse to theaxis 24′), but more generally may be tubular and of arbitrary shape incross section. As now described for the composite layer 108, all of thecomposite layers, which insulate portions of the conductor materialformed in the same or in different coil rows, have first and secondopposing end regions which, individually or collectively, are referredto as first and second coil ends 122, 124. The coil ends 122, 124 areformed about the coil aperture 102, shown in the cross sectional view ofFIG. 13F as having a circular shape. The core 104 is illustrated asextending beyond each of the ends 122, 124 for purposes of mountingduring the machining process, but it is to be understood that the coremay be terminated at or near the coil ends.

A feature of the coil 10 is that the layer 108 and other layers formedthereover include a shoulder region 126, alternately positioned at oneor the other of the coil ends 122, 124. In FIG. 13A, the shoulder region126 is adjacent the coil end 124. See, more generally, FIG. 13G whichillustrates, in a simple cross sectional view taken along a planeextending along the aperture 102, the series of insulator layers 108wherein, for each layer 108, a shoulder region 126 is formed at one end122 or at the other end 124 in an alternating pattern extending from theaperture 102 outward from the axis 24′.

The illustrated shoulder regions 126 are each in the form of a curvedcylindrical shape, positioned more or less symmetrically about the axis24′. The shoulder regions 126 may each have approximately twice thethickness of other portions of the layer 108 that extend along the majorouter surface 118. The shoulder regions may be formed by positioningapproximately twice as much composite material in the shoulder regionrelative to the other portions of the layer 108. The major outer surface118 and the surface 128 may be shaped by machining the layer 108 afterthe composite has cured. As illustrated, the shoulder region 126 may bedefined with an abrupt, step-like transition 129 between the twosurfaces 118 and 128 or the transition between the surfaces 118 and 128may be gradual, along a sloped surface formed between the surfaces 118and 128.

Referring next to FIG. 13B, a channel 130 along the path of a tiltedhelix is defined in the layer surface 118, creating a series of channelloops 132. In this example each of the loops 132 may be approximatelyelliptical in shape, it being understood that the individual loops arenot closed shapes because they are portions along a continuous helicalpattern. Collectively, the channel loops 32 define a path for placementof a segment of conductor which corresponds to a first row of coiledconductor. The loops may have more complex shapes than illustrated inorder to define or accommodate modulations and other variations in adesired conductor path. As illustrated, one or more of the loops 132 ofthe channel 130 may extend into the shoulder region 126. The channel 130is cut or otherwise formed in the cylindrical outer surface 118 so thatit extends a predetermined depth, d, into the layer 108 to define aconductor path. The actual depth of a portion of the channel, which isbelow the outer surface 118, may be equal to all or part of thethickness of the conductor to be placed therein, so that the conductormay be partly or entirely positioned within the channel 130.

The path defined by the channel 130 continues along the major surface118 into the shoulder region 126. Portions of the channel 130 formed inthe shoulder region 126 continue along a transition ramp that extendsfrom the shoulder surface 128 to a variable depth which effects acontinual transition in the channel 130, from a position at one level(e.g., a given depth below the surface 118) in the layer 108 to anotherlevel in the shoulder region 126 which corresponds to the intended depthof a yet-to-be-formed channel in a next of the layers 108 of insulatorto be placed about the layer 108 after conductor is positioned in thechannel 130.

A feature shown in FIG. 13C is the formation of two paths, i.e., achannel fork, wherein the channel 130 extends along two differentdirections. As the channel depth decreases in the shoulder region, thechannel bifurcates into a first path 131 that continues along thesurface 128 to a transition 129 and a second path 133 that continues ina direction away from the transition 129. With this arrangement, after aconductor is placed in the channel 130, the conductor may be positionedalong the second path 133 until the channel for the next coil row isformed.

Still referring to FIG. 13C, a winding process begins with positioning aspool 134 of conductor 138 at a first of the end regions 122. Thespooled conductor has a continuous length of sufficient distance, end toend, to turn conductor through all of the channel loops 132 of the coil10, thereby defining a series of conductor loops 136 in every one of theconcentric coil rows in a splice-free manner. In this regard, referenceto a conductor as splice-free means that, although a conductor segmentof given length can be formed of multiple, connected sub-segments, asplice-free conductor is one in which there are no discrete connectionseffecting continuity along the length. This is typically because theentire length of the conductor has been initially formed and thenpreserved as one body having an uninterrupted and continuous length. Byway of example, a filament may be extruded to at least the given length.A splice-free conductor is not one formed from multiple segments whichhave been electrically separate from one another prior or duringinstallation in a conductor assembly (such as the assembly 10) and thenhave been coupled together (e.g., such as by mechanical means or bysoldering or by welding) and thereby characterized by one or moredetectable junctions that provide for electrical continuity along thegiven length. Rather, a splice-free conductor segment of given length isformed as a single unitary body without requiring during formation ofthe assembly any connection among smaller lengths thereof to effectcontinuity. In the case of multifilament conductor, a splice-freemultifilament conductor segment of given length is also one which isformed as a single unitary body without requiring during formation ofthe assembly any connection among smaller lengths thereof to effectcontinuity. Notwithstanding the foregoing, the term segment, used in thecontext of a splice-free conductor of given length, may refer to one ormore portions of the length or the entire length.

A first end 142 of the conductor 138 is placed in a fixed manner nearthe end 122 of the layer 108 and a first segment 150 of the conductor138 is positioned in the channel 130. The conductor segment 150 is shownafter generating all of the loops 136 in the coil core row 146, i.e., afirst helical row of conductor loops 136 which is formed in the channel130 on the layer 108.

With the first segment 150 of conductor 138 placed in the channel 130,the conductor initially follows the second path 133 on the shoulderportion 128 with the spool 134 having been mounted on the core 104 atthe coil end 124. The placement of the conductor 138 in the path 133 andpositioning of the spool 134 on the core allow the conductor on thespool to remain attached to the conductor segment 150 while a nextcomposite layer is formed and tooled to generate another level ofchannel. By way of example, the core may be turned with the spoolattached thereto in order to shape a curved tubular surface of the nextcomposite layer and cut the channel. Once the next level of channel isformed, the conductor placed in the path 133 is removed and placed inthe first path 131 to continue the winding process along the nextchannel in a direction from the coil end 124 toward the coil end 122.

Placement of the conductor in the path 131 effects a 180 degree turn ofthe conductor 138 about the end 124 in order to position the conductorfor insertion in another channel in order to form a second coil row.Accordingly, FIG. 13D illustrates a layer 156 of composite materialformed over the core row 146 and layer 108, after having been cured andmachined to form a suitable curved tubular surface. The layer 156includes a machined outer surface 162 also having a curved tubular shapeinto which a second channel 166 is machined. The layer 156 furtherincludes a shoulder region 126 adjacent the coil end 122 and havingfeatures as described for the shoulder 126 which forms part of the layer108, i.e., being twice the thickness of the portion of the layer 156within the outer surface 162, and having a cylindrical outer shouldersurface 128. The channel 166 is formed, e.g., by machining, in thesurfaces 162 and 128 to define a second helical path for receiving asecond segment 152 of conductor.

When the layer 156 is turned, e.g., on a CNC machine, the spool 134 andassociated conductor 138, being attached to the core 104, turn with thelayer 156 as the channel 166 is machined therein. As described withregard to the layer 108, the shoulder region 126 of the layer 156 may bedefined with an abrupt, step-like transition 129 between the twosurfaces 162 and 128 or the transition between the surfaces 162 and 128may be gradual, along a sloped surface formed between the surfaces 162and 128. Also, as described for the shoulder of the layer 108, thechannel 166 includes a portion formed in the shoulder region, extendingfrom the surface 162 and extending toward the shoulder surface 128. Thechannel depth in the shoulder region 126, with respect to the surface128, ranges in depth as discussed with respect to FIG. 13C. The portionof the channel extends a variable depth below the outer surface 128 ofthe layer 156 up to a predetermined depth below the outer surface andmay continue along the surface 128 to the transition 129. Also asdescribed with respect to the layer 108, the portion of the channelalong the surface 128 includes two paths wherein the channel 166 extendsalong two different directions, one of the paths 131 continuing to thetransition 129 and the other path 133 continuing in a direction awayfrom the transition 129 so that the conductor may be initially placed inthe path 133 and then, after channels for the next coil row are formed,be placed in the path 131.

FIG. 13E illustrates the partially fabricated coil 10 having the segment152 of conductor 138 placed in the channel 166 to provide a secondhelical coil row 170. Both the channel 166 (see FIG. 13D) and the coilrow 170 are helical, with the channel 166 comprising loops 172 and therow 170 comprising loops 174 of conductor 138. The tilt angle of thechannel and conductor loops 172 and 174 is opposite the tilt angle ofthe channel and conductor loops 132 and 136 of the coil core row 146.Pre-definition, e.g., by machining the channel 166, of the coil path forthe second coil row 170, enables fixed placement of the conductorsegment 152 along the curved surface 62 of the tubular shaped layer 156.This arrangement avoids slippage and minimizes other forms of movementof the conductor length as it extends beyond the core row 146. Also, asseen in FIG. 13E, with the segment 152 wound along the channel 166, thespool 134 is next positioned on the core 104 adjacent the coil end 122,with a portion of the conductor 138 positioned in the path 133 while thenext layer of insulator is formed and a channel is formed therein.

A series of additional helical coil rows 12 are formed over the rows 146and 170. Initially with the conductor 138 extending from the segment 152at an end 122 of the layer 156, a first in a series of additionalinsulator layers 176 and a first in a series of additional coil rows 12are formed, and the alternating sequence proceeds in a manner similar tothat described for forming the initial sequence of the composite layer108, the coil core row 146, the composite layer 156 and the coil row170. The spool is alternately affixed to different coil ends 122, 124while each next insulator layer 176 is fabricated with a channeltherein. In other embodiments, the insulator layers 108, 156, 176, maybe pre-fabricated, with channels formed along the surfaces, andpositioned over the prior-positioned layers. The pre-fabricated layersmay be slid over one another or may be assembled from components having,for example, a clam-shell configuration, wherein each layer is formed oftwo components which, when placed together, form a curved tubular shapesuitable for the coil assembly of FIG. 8.

The described fabrication sequence enables formation of splice-freemagnetic coils in a helical, e.g., double helix, configuration around acurved axis. With this sequence it is no longer necessary to route theconductor from a lower insulative level radially upward to protrude outfrom the lower level in a region where the next insulative level is tobe formed. The fabrication sequences disclosed herein may be fullyautomated with conventional equipment such as a CNC machine. The abilityto build sequential coils rows with splice-free conductor addsreliability and reduces potential concerns relating to solder joints andcontact resistances.

The term “conductor” as used herein refers to a string-like piece orfilament of relatively rigid or flexible material, commonly referred toas cable or wire, being of the type comprising either a singleconductive strand or multiple ones of such strands grouped together asone functional conductive path. The term multi-strand conductor refersto such a conductor formed as a single identifiable unit and composed ofmultiple conductive strands which are twisted, woven, braided orintertwined with one another to form an identifiable single unit ofwire. Reference to one multi-strand conductor means application of thesingle identifiable unit as one functional unit and excludes havingmultiple ones of the individual functional units grouped togetherfunctionally when the multiple ones are not twisted, woven, braided orintertwined with one another. As used herein, multi-strand conductoronly refers to arrangements wherein the multiple strands are twisted,woven, braided or intertwined with one another to form the single unit.According to the invention, multi-strand conductor may take the form ofconductor that embodies a circular or a non-circular contour in crosssection.

Numerous cross sectional channel shapes and conductor shapes may be usedin constructing the coil 10. The conductor 138 may be a solid core or amulti-strand conductor, having a circular shape in cross section, asquare shape in cross section, a rectangular shape in cross section or arelatively flat profile, tape-like form. A multi-strand conductor havingA rectangular shape cross section may be a braided copper conductor or aRutherford type cable used for superconductor applications. Theconductor may be, for example, a YBCO-based high temperaturesuperconductor wire having a tape-like profile with a width dimension ina range, for example, between 2 mm and 5 mm, and a thickness in therange, for example, of 0.09 mm to 0.3 mm.

Generally, embodiments of the invention now provide a channel, such asone of the channels 130 or 166, in each of multiple conductor rows of acoil, having a profile suitable for accommodating a conductor of desiredcross sectional shape. Providing such a channel may result in one ormore additional benefits depending on the corresponding channel profile.

For example, with a conductor having a circular cross section, thechannel may have a corresponding circular shape with a width sized veryclose to or the same as the conductor diameter, and a depth ofapproximately one half the conductor diameter. With this arrangement,and a subsequent overcoat of another layer of composite, such as whereinone of the layers overcoats the conductor segment and portions of thelayer, it is possible to precisely define placement of the conductorsegments and constrain the segments from movement in the presence ofhigh magnetic fields. This placement can be totally independent ofconductor placement in an underlying coil row.

With reference to FIG. 14, there is shown an exemplary double helix pairof magnet coil rows 12C and 12D (having opposing tilt angles) followinga helical path about the axis 24′ and comprising, relative to the loops136 of FIG. 13, a higher frequency sinusoidal component, e.g., aquadrupole component, in each loop 202 thereof. Multiple ones of theillustrated rows 12C and 12D may be used in addition to multiple ones ofthe rows 12A and 12B to form a combined function magnet assembly forgenerating dipole and quadrupole fields, e.g., as needed for Fixed FieldAlternating Gradient (FFAG) accelerators. Embodiments of the inventionare not limited to the afore-described double helix configuration.

If needed, the double-helix geometry also allows manufacturing ofmagnets formed along curved axes with precisely designed combined fieldfunctions. For example, a design can superimpose over the same segmentof a curved axis both a dipole field for bending a beam trajectory and aquadrupole field for focusing the beam. The perspective view of FIG. 14shows a pair of coil rows 12C and 12D suitable for assembly in the coil10 (see, also, FIG. 13 F) wherein each row has a wiring pattern forgenerating a quadrupole field. As illustrated, the rows 12C and 12D areadjacent one another to create a double helix configuration about theaxis 24′.

The fabrication features described for the embodiment of FIG. 13 includeformation of channels 130 comprising individual channel loops 132. Aprocess is now described for machining such conductor channels in coilrows which follow a curved axis or for which the aperture radius variesas a function of position along an axis. The following exemplary processenables machining of such channels in a support structure or coreproviding a curved aperture having a constant radius of curvature and atotal bending angle of about 90 degree. The same process is applicableto other geometries, in particular those with changing aperture radiiand varying radii of curvature.

A machining process, suitable for incorporation with the fabricationprocess of FIG. 13 begins with provision of a curved support structurecorresponding to a partially fabricated coil assembly. The structure mayas now illustrated be of a shape conforming to the desired magnetgeometry. FIGS. 15A 15B and 15C illustrate in partial schematic views apartially formed coil assembly 210 mounted on a tooling machine duringan intermediate stage of manufacture to create a magnet coil assemblysuch as the coil 10. The partially formed coil assembly 210 comprisesone or more coil rows 12 (e.g., row 146 or row 170 of FIG. 13), one ofwhich is along an exposed surface shown in the figure. The coil rows 12may be formed, for example, in a resinous composite layer as describedin FIG. 13. According to other embodiments the assembly may befabricated with a Direct Helix manufacturing process such as describedherein. The assembly is mounted for rotation about the centerpoint 214of the curved axis of symmetry 24′. The center point coincides with astraight axis 220 of rotation on a CNC tooling machine 224. The assembly210 rotates around the axis 220 during a machining process which definesa channel for placement of conductor in accord with a predefined wiringpattern in each coil row. That is, first and second opposing ends 217and 219 are each fixedly secured to a chuck 221 or a tail stock 223. Thechuck and tail stock are aligned with the straight axis 220 while theends 217 and 219 are in offset positions relative to the chuck and tailstock so that the axis 24′ can rotate about the axis 220. With thisarrangement the rotation of the assembly 210 about the axis 220 permitsprecision machining along a curved surface. A feature of the inventionis that geometric irregularities along the surface 118 have minimal orno effect on the precision because the tooling machine can be programmedto generate a space curve relative to the predefined central axis 24′.For example, when fabricating such an intermediate structure, which maybe in the shape of a tube, or curved cylinder, formed about a centralaxis of symmetry, tolerances and errors in the fabrication processresult in topological variations along the surface. These variations aredepartures from the ideal symmetry of the surface about the centralaxis. By forming channels, e.g., for placement of conductor therein,based on spatial relationships between a defined axis and the desiredpositions of points along the channel, manufacturing precision forgenerating the channel at each point is independent of topologicalvariations along the surface. A channel is formed according to a seriesof points each having a predefined distance from an axis of symmetry.For points in the series, the predefined distance is a distancemeasurable along a plane transverse to the axis 24′ and passing throughthe point. Although numerous other techniques are suitable, forming ofthe channel may include application of a cutting tool such as a routerbit along the surface. The variable router position is based ondistances from an axis, which is not necessarily an axis of symmetry.Consequently for embodiments wherein the channel does not penetratethrough the structure being tooled (as is the case for some embodimentsfabricated according to the Direct Helix manufacturing process), theresulting position of the channel relative to the axis is primarilysubject to manufacturing tolerances associated with positioning of thetool relative to the axis. For such embodiments, channel position isindependent of topological variations along the surface.

The channel 130 may be may be machined in a resinous core support. Thisis followed by placement of conductor therein, and the sequencedescribed in FIG. 13 is repeated, including formation of a layerthereover which is machined to create another coil row for receipt ofanother level of conductor. FIG. 15A illustrates the assembly at a firstposition of rotation about the axis 220 showing a view wherein the axis24′ is in a plane parallel with the view. FIG. 15B illustrates theassembly at a second position after a ninety degree rotation about theaxis 220. FIG. 15C illustrates the assembly at a third position after aone hundred eighty degree rotation about the axis 220 relative to FIG.15A, wherein the axis 24′ is again in a plane parallel with the view.

The channel 130 may be cut with a high speed router bit 230 that ismounted for displacement in accord with a Cartesian (X,Z) coordinatesystem relative to the axis 220 for displacement about the axis 24′. Theassembly 210 rotates around the axis 220 which axis is parallel to the Xaxis such that the assembly 210 undergoes rotational displacementindependent of the router bit displacement. The router position isadjustable with respect to the surface in such a way that the tip of therouter bit follows the required channel pattern along the surface of thecoil form to create a channel of defined depth. The channel pattern isdetermined based on the transformations and optimizations that beginwith Equations 3 and result in a coil pattern for each of the rows 12.

The coordinated movement of the router bit 230 and the rotating assembly210 is based on a series of points generated for the coil patternwherein movement along each axis is independent. For example, with theassembly rotating about the axis 220, it is necessary to impart adisplacement along the X axis to follow the intended pattern along thelayer surface 118. However, movement of the rotating surface 118requires movement of the bit 230 along the Y and Z axes to assure propertracking of the bit during rotation. Further, accuracy of machining isenhanced by performing coordinated interpolations and movements inaccord therewith along the multiple axes to assure smooth continuousmovement of the bit along the surface 118 to replicate the modeledpattern resulting from the optimization process of FIG. 10.

In the embodiment of FIG. 15 for the example assembly 210, a rotationalmovement of the bit in the X-Z plane enables positioning of the bit sothat at any point it is always perpendicular to the tangent vector alongthe surface 118 at that point while the assembly 210 rotating. The toolhas rotational freedom in a plane to form the channel with a profilethat has a central axis which is always perpendicular to a tangentvector along the surface. As the direction of the tangent vector changesthe tool rotates accordingly. This degree of freedom can accommodatevaried channel shapes such as for receiving conductors having shapes incross section that are not circular, e.g., that might be rectangular.The tool can also vary from an angle perpendicular to the tangent vectoralong the surface in order to accommodate conductor shapes of varieddesign such as disclosed in U.S. application Ser. No. 12/061,813.

For the example shown in FIG. 15, when the router bit 230 rotates in theXZ plane any such movement affects the X and Z coordinate position ofthe bit and appropriate (X,Z) displacements must be imparted to the bitto offset these displacements and thereby achieve the required (X,Z)position relative to the axis 24′. In order to machine grooves into thebent surface of the assembly 210 with a required geometry, it istherefore necessary to supply the machine controller with pointcoordinates along each variable axis. The number of point coordinatesthat are needed for the machining process depends on the requiredprecision of the channel to be machined and can reach a million points,before interpolation, for a large coil such as an assembly having an arclength on the order of one meter and an aperture diameter of 10 cm. Forcoil designs wherein the axis of symmetry is not in a single plane, anadditional degree of freedom may be needed to control the angle of thebit 230 with respect to the layer surface.

FIG. 15D is a view in cross section showing an exemplary portion 285 ofthe channel 132 extending into the surface 118 of the assembly 210. Theview is along a plane transverse to the direction of the channel. Inthis embodiment a router bit 290 is positioned at a ninety degree anglerelative to a tangent vector 280 along the surface. The resultingchannel is formed about an axis of symmetry 295 which is orthogonal tothe surface 118. The channel portion 285 is shown extending a depth dbelow the surface 118. In other embodiments, the channel may include aflat lower surface and vertical side walls as described in Ser. No.12/061,813 and the axis of symmetry 295 may be at an angle with respectto the tangent vector to accommodate the conductor in accord with theteachings of Ser. No. 12/061,813. Notwithstanding such changes in theangle of the axis of symmetry 295, (wherein along the trajectory of therouter bit 290, the bit is positioned at a variable angle relative tothe aforedescribed ninety degree angle), the channel is also formedaccording to a series of points each having a predefined distance froman axis of symmetry. As described for the illustration of FIG. 15, thepredefined distance is a distance measurable along a plane transverse tothe axis, e.g., axis 24′, and passing through the point. Althoughnumerous other techniques are suitable, forming of the channel mayinclude application of a cutting tool such as a router bit along thesurface. The variable router position is based on distances from anaxis, which is not necessarily an axis of symmetry. Consequently forembodiments wherein the channel does not penetrate through the structurebeing tooled (as is the case for some embodiments fabricated accordingto the Direct Helix manufacturing process), the resulting position ofthe channel relative to the axis is primarily subject to manufacturingtolerances associated with positioning of the tool relative to the axis.That is, for such embodiments, channel position is independent oftopological variations along the surface. When the conductor placed inthe channel is not of a circular shape in cross section, e.g.,rectangular, with the axis of symmetry 295 having a variable angle withrespect to the tangent vector, the series of points having predefineddistances from an axis, e.g., like axis 24′, may be a function of theconductor shape and the variable angle. That is, the pattern conformswith necessary displacements from the axis to assure generation ofdesired multipole orders and suppression of undesired multipole orders.

With application of the optimization procedure of FIG. 10, as well asother optimization procedures described herein, it becomes possible toreduce the magnitude of systematic errors, e.g., removal of undesiredmultipole components, to the extent that the predominant source of erroris error which stems from manufacturing tolerances, referred to hereinas random error. As discussed above, such manufacturing tolerances canbe reduced by tooling the channels relative to an axis so thatpositioning of the channel and positioning of the resulting conductorare primarily subject to manufacturing tolerances associated withpositioning of a cutting tool relative to the axis. With this approach,manufacturing tolerances are independent from, i.e., not a function of,topological variations along the surface. In illustrated examples, witha channel formed through a surface and into a structure, the channelcontour is derived directly from a wiring pattern based on distancesmeasured from a central axis.

Consequently, using an optimization procedure such as described in FIG.10 during a manufacturing process such as illustrated in FIGS. 13 and15, systematic errors can be reduced to levels smaller than the errorsassociated with manufacturing tolerances. Further, by tooling a channelbased on displacement relative to an axis, in accord with the examplesof FIG. 15, manufacturing tolerances can be achieved on the order of0.01 mm. Tolerances on the order of 0.001 mm or less also may beachievable.

Still, it is possible that manufacturing variations will introducerandom errors that can result in generation of multipole componentshaving undesirably large influences relative to the suppressedsystematic errors. This is especially true when an optimizationprocedure is employed to remove systematic errors such thatmanufacturing tolerances are the predominant source of error. Accordingto another set of embodiments, applicable in applications requiringgreater field uniformities, a procedure suppresses such unwanted termsas they occur during manufacture. An example process, in accord with theflow chart of FIG. 18, can be employed during the fabrication processesshown in FIGS. 13 and 15. Manufacturing errors in a complete conductorassembly, comprising multiple coil rows, are an accumulation of errorsintroduced during the fabrication of each coil row. Recognizing this,once one or more conductive coil rows 12 are formed, a field calculationis performed for the partially fabricated structure. For example, acurrent may be run through the one or more coil rows 12 to measure fieldmagnitudes and characterize the field quality generated by the limitednumber of the rows in the partially fabricated assembly.

If multipole field components of unwanted magnitude are detected, thesecan be offset by introducing modulations in the next layers using thedescribed method. That is, the deviation between calculated and measuredfield values are determined and, as described in greater detail in FIG.10, optimized values are determined for a set of parameters which offsetthe unwanted magnitudes. The process of measuring deviations in fullyformed coil rows 12 and introducing offsetting corrections in anoverlying layer may be repeated as the assembly proceeds with formationof the additional coil rows 12. A feature of this process is that theeffects of manufacturing tolerances on field quality decrease as afunction of radial distance. Assuming the manufacturing tolerance is thesame for all coil rows 12 ranging from an inner most coil row to anoutermost coil row, errors associated with outer coil rows will haveless influence on field quality in the aperture region than will errorsassociated with outer coil rows.

A fabrication method which suppresses generation of unwanted multipolefield orders begins with determining a winding configuration that meetsthe design specifications of the fully fabricated conductor assembly,e.g., magnet coil. This includes a calculation of skew and normalmultipole components as shown in Table 1. Next, one or more coil rows 12are formed. These may be multiple pairs of double helix coil rows. Forthe process shown in FIGS. 13 and 15, fabrication includes machining asupport channel for each coil row in a layer of composite material ofthe coil support structure. The structure may be curved as illustratedor in the form of a straight cylinder, or may include flared regions asdescribed hereafter for still other embodiments. The conductor is placedin the machined channel, and the process shown in FIGS. 13 and 15 isrepeated to create a desired number of coil rows, fewer than the totalnumber of coil rows in the completed assembly. With this partiallyfabricated structure, measurements are made of the multipole content.The measurement can be done with well known techniques, e.g. using arotating pickup coil in the aperture of the coil. A Fourier analysis ofthe voltage induced in the rotating pickup coil allows calculation ofthe multipole content of the fabricated coil rows. Such techniques canprovide accuracies of measurement for higher-order multipole fields onthe order of 10⁻⁵ relative to the main multipole component.

With the measured data, deviations are determined, for each multipoleorder, between measured and required multipole content (e.g., in termsof magnitude) of the magnet under construction. With this data wiringpatterns for one or more of the next coil rows are calculated in amanner which compensates for the measured deviations from the requiredfield configuration.

The conductor layout for these coil rows can be determined as describedin FIG. 10 except that when the assembly is performed along a straightaxis, no bending transformation is performed: Generate each helicalpattern in accord with the functions X(θ), Y(θ) and Z(θ); calculate themultipole content of the winding configuration; generate an objectivefunction, e.g.,

${\chi^{2}\left( {ɛ_{2},\ldots\mspace{11mu},ɛ_{n}} \right)} = {\sum\limits_{k = 2}^{k\;\max}\left\lbrack {{A_{k}^{2}\left( {ɛ_{2},\ldots\mspace{11mu},ɛ_{n}} \right)} + {B_{k}^{2}\left( {ɛ_{2},\ldots\mspace{11mu},ɛ_{n}} \right)}} \right\rbrack}$

which depends on the unwanted multipole field components, i.e., based ondetermined deviations, and which reaches a minimum when the deviationsreach desired minimum values; and modifying parameters to incorporatevalues of ∈₂, . . . ∈_(n) which result when the function reaches aminimum. With respect to this process and the process of FIG. 10 it maynot be necessary for the minimum reached by the objective function to bean absolute minimum. This procedure determines modulation components(amplitudes and phase angles) that are introduced into the wiringpatterns for the one or more of the next coil rows to offset measureddeviations from the required field configuration. As manufacturingcontinues one or more additional ones of the total number of coil rowsare formed about the support structure to continue fabrication of theassembly. Also as manufacturing continues the same process ofmeasurement, determination of deviations and calculation of offsettingmodulation components may be repeated. Such a process of providingadjustments in outer coil rows to compensate for manufacturing errors ininner coil rows significantly decreases deviations of higher-ordermultipole fields of the complete coil from their design goal. Theprocess may provide consecutive adjustments. The process converges tohigh field fidelity since higher-order multipole fields, i.e.quadrupole, sextupole, etc., decrease according to (R/R_(o))^(n−1),where r is the radius of the conductor, R_(o) is the reference radiusand n is the multipole order. This factor increases rapidly for theouter rows and a given reference radius. Placement accuracy is thereforeless important for the outer coil rows, which are farther way from thereference radius and conductor placement accuracy is less critical forthe outer coil rows.

In the partial view of FIG. 12A, a coil magnet 30 is formed along astraight axis X consisting of multiple coil rows 32 (only one of whichis shown) in accord with equations 3 and n=1, the row having anexemplary 100 turns and a coil aperture radius of 25 mm. FIG. 12Billustrates the magnet 30 in a cross section taken along a planetransverse to a point along the axis X, illustrating that the magnet 30is an assembly comprising an arbitrary number of coil rows 12 _(i)formed about one another and an aperture 102. A feature of the magnet 30is that if a field is generated in the magnet there will be a net axialfield component. This is to be compared with a double helix design inwhich coil rows are provided with opposing tilt angles to cancel axialfields and maximize fields in planes transverse to the axis of symmetry.According to the embodiment of FIG. 12 each of the coils is configuredwith values of A_(n) which impart a net axial field component in thesame direction along the axis X so that the net axial field component ofthe magnet 30 is the sum of the magnitude of the axial field componentgenerated by each coil row 12 _(i). In the example embodiment, all ofthe A_(n) values have the same sign and therefore each coil row “tilts”in the same direction with respect to a plane transverse to the axis.That is, with reference to FIG. 13C, all coil rows exhibit a tilt in thesame direction as shown for the conductor loops 136, instead ofalternate rows exhibiting an opposing tilt as shown for the loops 174 ofFIG. 13E. See also FIG. 12C, a partial schematic view showing only oneexemplary conductor loop 34 _(i) in each of three consecutive coil rows32 _(i) (i=4, 5, 6). Each loop 32 _(i) in each row is representative ofthe series of loops in that coil row 32 _(i), having a tilt angle“a_(i)” in the same direction with respect to a plane P_(i) transverseto the axis X. Individual loops in each coil row are shown at differentpositions along the axis X for purposes of clarity. The illustratedloops of different coil rows shown in FIG. 12C are each configured toprovide a main dipole field, but it is to be understood that otherembodiments may incorporate multiple multipole configurations withinindividual coil rows or among different coil rows. In one embodiment thecoil rows of the magnet 30 may have essentially the same helicalconfiguration in accord with the X( ) equation, i.e., a₁=a₂=a₃. In otherembodiments the individual coil rows can be designed to providedifferent performance features and can be independently controlled. Forexample, the tilt angles a_(i) may differ from row to row to developvariations in proportions of axial and transverse field strength. Asshown in FIG. 12C this may be effected by separately connecting thedifferent coil rows to different power supplies PS_(i) for independentcontrol. In another design the magnet 30 may comprise multiple groups ofcoil rows wherein each group is connected to a separate power supply. Inthese embodiments the power supplies can be controlled to proportion therelative field strength of the magnet 30. This not only contemplatesproportioning axial and transverse field strengths, but also relativestrengths of different multipole orders. In other embodiments of coilassemblies formed with multiple coil rows (including but not limited todouble helix configurations) it is also advantageous to separatelycontrol individual coil rows or groups of coil rows. For example, in aparticle beam application, changing the focusing capability in a magnetassembly can be effected by modifying the current supplied to differentwinding configurations, e.g., dipole and quadrupole.

The multipole content of a given coil geometry is strongly affected bythe symmetry of the winding configuration. In the ends of a single-layerhelical winding, such as the illustrated row 32, the up-down symmetry isbroken because all of the tilts in different rows are in the samedirection. In double helix embodiments the asymmetry of each coil row iscountered by another coil row with the formation of coil pairs havingequal but opposing tilt angles. For the magnet 30 having a dipole fieldas the main component, if left uncompensated, this asymmetry wouldintroduce higher-order multipole fields near the ends of the coil. Ascan be seen from Table 5, even in the center of the coil, about 3 coilaperture diameters away from the ends, a significant quadrupole field B₂is present relative to the double-helix coil magnet 16 of FIG. 11. Acomparison with Table 1d, which illustrates field components for thesame coil geometry, but with the two rows 18A and 18B (providing adouble helix configuration wherein coils in the pair have opposing tiltangles), shows this effect. However, by applying an optimizationprocedure such as described in FIG. 10 for the bent coil of FIG. 8, thehigher-order fields can be minimized for magnets formed with a straighthelical coil row such as the row 32. In other embodiments the coil rowsmay be formed along a curved axis. In this example of a straightgeometry, the procedure of FIG. 10 is modified to not include the stepof performing the bending transformation. The results of thisoptimization for the straight geometry coil magnet 30 are shown in Table6, where again only the quadrupole, sextupole and octupole componentshave been optimized. The multiplication factors obtained from theoptimization are given in Table 7. As a result of performing theoptimization, modulations are introduced into one or more coil rows tooffset the magnitudes of higher order field components associated witheach of the coil rows 32. For example, the quadrupole field B₂ isreduced to a value less than 10⁻⁹ that of the main filed dipolecomponent. Other components which were simultaneously optimized exhibitsimilar reductions in field strength. The calculations confirm thathelical coil rows having high quality fields can be constructed withouta double helix configuration as described in the '042 patent. Suchmagnet designs can provide an axial field in combination with a highquality transverse field, e.g., in particle beam accelerators. The sameconcepts can be applied in applications of beam bending and focusing.The main field component may be a quadrupole field or an even higherorder field. Field quality may be determined as described for otherembodiments, based on multipole content in a circle along a plavetransverse to the axis.

TABLE 5 Multipole content of a straight single layer coil in center ofwinding. In comparison to the double-layer coil (see Table 1d) anincreased quadrupole component B₂ is introduced. The coil current hasbeen adjusted to 472 A generating a dipole field of 1000 Gauss. MP OrderAn Bn Cn 1 2.02E−03 1.00E+03 1.00E+03 2 −1.13E−03 −1.88E−02 1.88E−02 3−3.42E−05 −1.76E−03 1.76E−03 4 −2.01E−06 1.65E−03 1.65E−03 5 7.55E−07−1.67E−03 1.67E−03 6 4.44E−07 8.87E−04 8.87E−04 7 2.69E−06 9.89E−049.89E−04 8 2.02E−06 −2.19E−03 2.19E−03 9 −5.05E−06 −6.34E−04 6.34E−04 10−2.30E−06 3.05E−03 3.05E−03

TABLE 6 The multipole content of the straight, single-layer coil at areference radius of 20 mm (80% of coil aperture) after optimization ofquadrupole, sextupole and octupole components. The current remained at472 A. In particular the quadrupole field component B2 has beensignificantly reduced. Compare to Table 5. Optimizing Multipole Fieldsin Single-Layer Straight Coil MP Order An Bn Cn 1 2.02E−03 1.00E+031.00E+03 2 −1.85E−06 2.94E−07 1.87E−06 3 −4.16E−05 −2.72E−07 4.16E−05 44.15E−05 9.26E−07 4.15E−05 5 7.46E−07 −1.36E−03 1.36E−03 6 −8.27E−088.85E−04 8.85E−04 7 2.71E−06 6.67E−04 6.67E−04 8 2.48E−06 −2.20E−032.20E−03 9 −5.10E−06 −9.70E−04 9.70E−04 10 −2.56E−06 3.04E−03 3.04E−03

TABLE 7 Optimized amplitude factors and phase angles for straight,single-layer coil. ε_(n) Δφ_(n) ε₂ sin (2θ + Δφ₂) 1.189359E−05  5.999469E−02 ε₃ sin (3θ + Δφ₃) 8.332971E−07 −4.671344E−03 ε₄ sin (4θ +Δφ₄) −8.235957E−07   −2.583852E−02

As shown above for the double helix coil magnet 16 of FIG. 11,higher-order multipole fields are highly suppressed in the center ofdouble-helix coils at a distance from the coil ends of 2 to 3 times thecoil aperture. The 300 mm magnet 16 has for the inner row 18A first andsecond coil ends 22A and 22B, and has for the outer coil row 18B firstand second coil ends 22C and 22D. With reference to FIG. 16A, atpositions approaching the coil ends (+/−150 mm) the higher-order termsrise significantly beginning at about 50 mm (or approximately oneaperture diameter) from the coil ends, reaching a maximum value andfinally decreasing to zero, typically within one aperture diameterbeyond the coil ends. For most applications performance specificationsare based on relative magnitudes of components, such as by performing anintegration of each higher order multipole field over the full length ofthe magnet and comparing the value to that of the main field component.This way a determination is made as to whether the higher-ordermultipole fields, resulting from the coil ends, are acceptable.

According to embodiments of the invention, higher field uniformity isachievable over the full axial length of the magnet coil and,specifically, in regions within one aperture diameter of the coil ends.A first design providing this improved field quality is based onintroduction of variable aperture and coil row diameters along themagnet axis, e.g., within one aperture diameter of each coil end of acoil row. The term flared region as used to describe such embodimentsmeans a geometry which includes a curved contour extending radiallyoutward from the axis around which a helical conductor pattern extends,and varying in distance from the axis as a function of position alongthe axis. In example embodiments the curved contour is symmetric aboutthe axis but the invention is not so limited.

FIG. 17A shows a coil assembly 40, which is a modification of the coilmagnet 16 wherein each of two coil rows, inner coil row 42A and outercoil row 42B, together form a double helix coil. The coil rows 42A and42B each include a flared region 48 along each of the coil ends. Theinner coil row 42A has opposing coil ends 44A and 44B while the outercoil row 42B has opposing coil ends 44C and 44D. The coil rows otherwiseconform to the geometry described for the coil rows 18A and 18B of FIG.11. That is, there are 100 turns in each coil row and along the innerportions of the axis (i.e., more than one 100 mm aperture diameter fromeach coil end) the aperture radius is 25 mm and the radius of the outercoil 42B is 27 mm. FIG. 17B is an elevation view of the coil assembly 40illustrating positioning about a central, symmetric axis 46.

As shown in FIG. 17, within 100 mm from each coil end 44A and 44B, thecoil aperture radius (also corresponding to the radius of the inner coilrow 42A) increases monotonically. In a center region extending inward100 mm from each coil end 44A and 44B, both the coil aperture radius andthe radius of the inner coil row 42A are 25 mm, corresponding to a 50 mmdiameter. Within 100 mm from each coil end 44C and 44D, the radius ofthe outer coil row 42B increases monotonically. At 100 mm from each coilend 44C and 44D, the radius of the outer coil row 42B is 25 mm. At thecoil ends (44A, 44C) and (44B, 44D), the aperture radius increases toabout 50 mm or about a 100 mm diameter at the coil ends. The radius ofthe outer coil row 42B increases monotonically from a 27 mm radius (54mm diameter) to a 54 mm aperture radius (108 mm diameter) at the coilends. In other embodiments, the assembly may include a central flaredsection of variable aperture (or coil) radius formed between sections ofconstant aperture (or coil row) radius. See, for example, the sections355, 360 and 370 of magnet assembly 300 shown in FIG. 23.

The manufacturing technology described in Ser. No. 12/061,813 enablesfabrication of flared regions 48 resulting in flared apertures 50. Alsoas described therein, precise support grooves are machined into asupport structure, in which the conductor is placed. The machiningprocess for the changing aperture is similar to what is performed formanufacturing of bent coils as described with reference to FIGS. 13 and15.

Generally, for fabrication of magnets in accord with the flared geometryof FIG. 17 a core is formed from a mold, over which a fiber reinforcedcomposite lay-up is formed to create a substrate in which grooves forreceiving coil can be formed. The core includes a slope corresponding tothe desired slope of the flared regions 48 and the core may be removedafter the composite lay-up is cured and machined. The core may beremoved with a solvent, e.g., water, or chemically. It is to beunderstood that multiple layers can be formed one over another to createmultiple double helix configurations or single helix configurations (seeFIG. 12) which may also incorporate flared end regions 48. That is, aseries of resinous composite layers are sequentially formed over oneanother with the intervening coils placed in grooves machined along eachlayer surface. See again Ser. No. 12/061,813.

The coil rows 42A and 42B each depart from a straight cylindricalpattern of constant radius (e.g., 25 mm or 27 mm) beginning at adisplacement of +/−50 mm from the center point CP along the axis 46. Theflared regions 48 of the coil rows may have, as illustrated, an abrupttransition to a linear slope of 0.25 at +/−50 mm, or may be formed witha gradual continuous transition to the constant slope. The resultingaperture radius at each coil end is 50 mm. Alternately, the flare mayfollow a quadratic slope (not illustrated) and other slopes may also befollowed to provide an optimum reduction in end field effects within theoriginal aperture radius. By providing a non-linear slope it is possibleto create a more abrupt decrease in the main field, e.g., dipole,component while also suppressing the higher order components whichnormally persist near the coil ends.

To illustrate improvements achievable by incorporating flared coil endregions 48 such as shown in FIG. 17, FIG. 16A illustrates the sextupolecomponent which is prevalent for the straight geometry configuration ofthe magnet 16 shown in FIG. 11. Notably at a reference radius of 20 mmthere are peak sextupole components at about 125 mm and 175 mm from thecenter point CP of the magnet 16. FIG. 16 B illustrates the suppressedsextupole component realized at these same positions when the magnet 16is modified to incorporate flared end regions 48 according to FIG. 17.That is, for the magnet 40, the peak sextupole components near the coilends, i.e., at about 125 mm and 175 mm from the center point CP′,diminish by more than a factor of 10. For higher order multipole termsthe flare provides even greater suppression about the coil ends. SeeTables 8A and 8B which provide values of several different multipolecomponents in a plane transverse to the axis. Table 8A provides thefield values at +125 mm from the center point CP of the magnet 16, i.e.,per FIGS. 11 and 16A. Table 8B provides the field values at 125 mm fromthe center point CP′ of the magnet 40, i.e., per FIGS. 17 and 16 B.Table 8C illustrates that with incorporation of flared aperturegeometries of FIG. 17 the field at the center point CP′ of the magnet 40is not degraded. With the main field being a dipole field, the flaredend reduces the higher order sextupole field component to a relativelysmall value. When the main field component is a quadrupole field, theflare reduces the higher order field components to relatively smallvalues as well.

TABLE 8A Straight Ends MP Order An Bn Cn 1 −2.96E+00 9.23E+02 9.23E+02 21.76E+01 −7.49E−01 1.77E+01 3 6.37E+00 1.11E+02 1.12E+02 4 1.83E+01−1.78E+00 1.84E+01 5 7.57E−01 5.27E+00 5.32E+00 6 2.29E+00 −4.68E−012.34E+00 7 −1.14E+00 −7.87E+00 7.96E+00 8 −3.18E+00 5.16E−01 3.22E+00 9−3.79E−01 −1.83E+00 1.87E+00 10 −1.42E+00 3.41E−01 1.46E+00

TABLE 8B Flared Ends MP Order An Bn Cn 1 −1.84E+00 4.62E+02 4.62E+02 24.20E+00 −1.79E−01 4.20E+00 3 5.33E−01 8.08E+00 8.10E+00 4 2.16E−01−3.79E−02 2.19E−01 5 3.10E−02 8.88E−02 9.40E−02 6 7.74E−05 −3.28E−033.28E−03 7 7.83E−04 −1.04E−02 1.05E−02 8 −3.10E−04 −3.30E−04 4.53E−04 99.98E−05 −6.02E−04 6.10E−04 10 4.73E−05 −5.79E−05 7.47E−05

TABLE 8C Multipole content of straight coil with flared end. Nosignificant effect on multipole fields in the center of the coil whichis 3 coil apertures away from both ends. Multipole fields in the centerof the coil in axial directions. Current adjusted to 238 A generating adipole field of 1000 Gauss. The table shows high field uniformity in thecoil center, where all higher order fields are less or equal to a fewper mille relative to the main dipole field. Flared Ends - Center ofCoil MP Order An Bn Cn 1 1.15E−01 1.00E+03 1.00E+03 2 −8.64E−03−2.53E−03 9.00E−03 3 −7.80E−05 −2.05E−03 2.05E−03 4 1.43E−05 8.31E−048.31E−04 5 8.59E−07 −8.41E−04 8.41E−04 6 −4.74E−07 4.47E−04 4.47E−04 75.68E−08 4.98E−04 4.98E−04 8 1.78E−08 −1.11E−03 1.11E−03 9 −1.54E−10−3.17E−04 3.17E−04 10 −6.35E−10 1.54E−03 1.54E−03

Another embodiment for improving the field uniformity in the coil endsis based on the optimization procedure shown for bent coils. Theiterative optimization process is described in FIG. 19. This procedureis a generalization of the algorithms applicable to the bent geometry ofFIG. 8 and the straight coil of FIG. 12. The amplitude factors ∈_(n) areno longer constant but have to be adjusted depending on the axialposition and therefore depend on the azimuth angle θ. To accommodatethis axial dependence, the ∈_(n) values become functions of the angle θ.In one embodiment of the procedure each multiplication factor ∈_(n) canbe described as a power series and is then given by typically 2 to 4parameters as shown in the following equation.

$ɛ_{n} = {{\sum\limits_{n = 0}^{m\;\max}{a_{m}\theta^{m}}} = {a_{0} + {a_{1} \cdot \theta} + {a_{2} \cdot \theta^{2}} + \ldots}}$

The number of parameters that need to be optimized increases, but theiterative optimization is performed in the same way. A completerepresentation of the applicable algorithm is shown in FIG. 19. With thealgorithm modulation components are determined which when added to theX(θ) function, reduce the higher order multipole field components (e.g.,the sextupole component shown in FIG. 16A). As discussed for embodimentswhich incorporate flared coil end regions, embodiments which suppresshigher order multiploes near the coil ends in accord with anoptimization process, generally reduce the undesired orders by a factorof 102 or more relative to the main field component. When the main fieldcomponent is a dipole field, the unmodified end field has a significantasextupole component; and when the main field component is a quadrupolefield, the unmodified end field includes one or more significant higherorder components.

Embodiments of the invention may be fabricated in accord with a designreferred to as a Direct Helix (DH) Design and a process referred to as aDirect Helix (DH) Process. The Direct Helix Process enables one todirectly create a continuous conductor path along a tubular shapedstructure having a conductive outer surface. The tubular shapedstructure may be in the shape of a regular cylinder or may take anon-linear shape including the shapes illustrated in various ones of thefigures, including curved geometries along a curved axis and apertureshaving varied radii. In one series of embodiments, a continuoushelically-shaped conductor has varying material widths (measurableacross cross sections taken along planes transverse to the conductorpath) which can reduce the total resistance of the conductor while stillmaintaining desired magnetic field characteristics. The conductor crosssections can be adjusted and optimized to provide desired fieldcharacteristics and electrical properties. The conductive outer surfacemay be a layer formed on a tubular substrate or may be the surface of asolid, monolithic conductive tube formed, for example, of extrudedcopper or may be a metallic casting. The thickness of the outerconductor surface is not limited and certainly can range at least frommicrons to multiple centimeters.

The examples of design and manufacturing methods now described involvean electrically conducting tube positioned about a substrate andmachining away portions of the conducting tube to leave a continuousconductor path. The path may be in the form of a tilted helix formedalong the shape of a regular cylinder, but other multipoles andcombinations of multipoles are contemplated. In multi-layer coilembodiments, for each layer or coil row, the conductive coil pattern isformed along a surface that may be bonded or otherwise attached to alayer of insulator which may provide the function of a stabilizingsubstrate.

Generally, a desired conductor profile is formed in a tubular surface byany of numerous known techniques such as machining with a tool, etchingor laser cutting. All conductive material in electrical contact with theconductive surface, in regions outside of the defined conductor path, isremoved, leaving a void which may simply provide a spatial gap betweenloops of the coil, or which may be filled with suitable dielectricmaterial. In some embodiments, the voids can be filled with epoxy toprovide a desired mechanical strength and dielectric property or may beused as one or more cooling channels, e.g., for flow of water or liquidnitrogen along the surface of the conductor. The coolant may be indirect contact with each conductor. Further, the level of cooling can beimproved by introducing gaps between conductor layers. This reducescoolant flow resistance and more heat can be removed.

Embodiments of the invention may incorporate double helix windingconfigurations based in part on concepts described in the '042 patent,but winding geometries may vary from turn-to-turn and fromlayer-to-layer to achieve desired field configurations and field qualitycharacteristics analogous relative to those having been provided withprior-known “wire” wound coils. A larger number of choices of conductivematerials are now conceivable with embodiments of the invention, theseincluding copper, aluminum and numerous types of superconductingmaterials. Very robust coil windings are made available. Generally, manyconductive materials that do not lend themselves to conventional wiremanufacture are available to practice the present invention. Forexample, the invention allows the use of superconducting materials inthin sheets or tube shapes. In other embodiments high temperaturesuperconductors like YBCO can be used in the invented process bydirectly depositing layers of the material on to an appropriatesubstrate material as used in the manufacturing of tape conductors ofthe same superconductor. In such applications multi-layered coils can bemanufactured with a very small radial build-up, e.g., minimum coildiameter, since the conductor layers of superconductors like YBCO aretypically only 1 or 2 microns thick. Such embodiments are useful forhigh temperature superconductors which are of a brittle nature andhaving limitations on achievable bending radii.

Also, because the conductive coils may be formed in-situ with a materialremoval process, the invention allows for accommodation of very “large”conductors, i.e., having large cross sections, without encountering manyof the difficulties which might result from conforming a wire into ahelical pattern. On the other hand, very small and fine line geometriesfor coil configurations can be attained via, for example, an etching orlaser removal process. Thus embodiments of the invention are well-suitedfor medical devices and small sensors. Examples include magneticresonance imaging applications and catheters. Further, the inventionallows provision of variable conductor cross section along each turn orloop in a helical pattern to further reduce resistance, or to optimizefield shape. The invention is not limited to forming helical coil shapesabout an axis of symmetry and may be applied to create numerousconventional geometries along a surface by removal of material. Afeature of the invention is enablement of conductive patterns havingvery small radii of curvature otherwise not attainable with conventionalwire winding techniques. Many of the embodiments described herein may befabricated as Direct Helix designs and the method described in FIG. 15may be adapted to perform the material removal process.

A Direct Helix manufacturing process begins with provision of anelectrically conductive tube or layer that is bonded or deposited onto asupport structure. A groove, fully penetrating through the conductivematerial is cut into the layer such that a conductive path along thesurface remains, which forms a winding suitable for generating amagnetic field or which, in the presence of a changing magnetic field,induces a voltage. The groove cut into the conductive material leaves avoid or space which electrically isolates adjacent winding turns fromone another.

Multi-layered coil configurations are formed by combining such tubes orlayers in a concentric configuration with the coil rows insulated fromeach other, although the conductor forming each coil row may beelectrically wired in series to conductor in other rows to create amulti-level magnetic system. That is, coil ends formed along each tubeor layer can be connected to coil ends in one or more other tubes suchthat a continuous conductor path results for the multi-layeredstructure. In such an embodiment gaps can be introduced between themultitude of tubes, or layers of coil rows, which allow coolant to makecontact with multiple sides of the conductor for highly effectiveremoval of heat generated by the conductor.

An example of a coil configuration according to the invention, referredto as Direct Coil, and an associated, exemplary design process isdescribed for a dipole coil. The following description is limited to asingle layer coil or coil row, since the process of forming additionallayers follows the same procedure. The exemplary Direct Coils arehelical in shape and the configuration is referred to herein as a DirectHelix, or DH.

As for conventional coil designs we start with given specifications forthe dipole coil. Relevant parameters needed for the design of the dipolecoil are shown in Table 9:

TABLE 9 Typical design parameters for a dipole coil Parameter Unit ValueCoil aperture radius, R mm  50 Coil length mm 300 Nominal field strengthTesla as high as possible Nominal current A as high as possible Fielduniformity (relative to as high as Dipole) possible

For a given coil aperture and coil length, it is often desirable toattain the highest possible field strength in a continuous, normalconducting operation. In any magnet coil the achievable field strengthis limited by the amount of current that can be applied to the coilwithout overheating the windings or, in case of superconductors, withoutexceeding the critical current. For normal conducting coils it istherefore important to have a low resistance and a highly efficientcooling scheme. An initial design is performed based on thespecifications for geometric coil dimensions, nominal field strength andfield uniformity. With a requirement for a highly uniform transversefield, the coil design may be based on a double-helix coilconfiguration.

In one embodiment according to the invention, a coil geometry provides atilted helical winding pattern with features of lower resistance, moreefficient cooling and higher achievable field strength. The design ofthe DH coil may be accomplished as follows:

We start by defining a tool path with the space curve of Equation 2 orEquation 3 along which a router bit with a given diameter cuts a fullypenetrating groove, G, into a conductive layer having a tubular shape.In the current example the layer is in the form of a self-supportingaluminum or copper tube, but may be a coating provided on a tube-shapedstructure. The inner diameter of this tube is equal to the required coilaperture defined in Table 1. The machined groove provides a space, alsoreferred to as an insulative groove, between the turns of the helicalwinding pattern that is generated. The width, W_(G), of the insulativegroove, which is the distance between neighboring winding turns, isgiven by the cutting width, e.g., length or diameter, of the router bit.In the following example, the router bit is of a shape having acharacteristic cutting diameter and corresponding cutting radius. Once ahelical groove is formed, by removal of conductor material, a helicalconductor pattern remains.

Merely cutting a helical groove into a conductor does not result in asufficient conductor path to create a magnetic coil. As shown in theschematic view of an unrolled coil pattern of FIG. 12, additionalmachined grooves, labeled “Line-in a”, “Line-in b” “Line-out a” and“Line-out b” are needed to form lead-in and lead-out connectors andcomplete a continuous current path that form current entry and exitterminals to the winding.

Pairs of dashed lines shown in the unrolled view of FIG. 20A representouter edges of the router bit machining paths, MP, and the insulativevoid which results after the conductive material is removed. Theremaining strips, S, of conductive material, indicated in FIG. 20A formthe resulting helical-shaped conductor path. Of course the paths MP maybe formed by other methods such as etching.

To perform a field calculation and to estimate the resistance of theconductive strips, a mathematical description of the strips S isprovided. The strips may have relatively large widths, Ws, resulting ina ribbon-like shape of relatively high width-to-thickness ratio, or anapproximate rectangular shape with a lower width-to-thickness ratio. Asillustrated the strips may be open loops of elliptical shape or may bemore complex modulations, and generally, the strip widths may vary as afunction of the azimuth angle, θ.

Due to the approximate rectangular shape, in cross section, of thestrips, S, it may not be sufficient to calculate the resulting magneticfield using those approximations which have been suitable for aconductor having a circular shape in cross section. In lieu ofcalculating the resulting magnetic field, e.g., with a single infinitelythin filament that is centrally located within the strip, a morecomplete design method is provided for both modeling and modifying thepattern of the strips. The method incorporates optimization proceduresto achieve desired performance criteria for field uniformity, coilresistance and other parameters of interest.

FIG. 21 illustrates each strip, corresponding to an open,elliptical-shaped loop in the illustrated coil row, CR, may be describedby 4 curves, C₁, C₂, C₃, C₄, each spatially positioned along one of thecorners of a strip S. That is, assuming the strip has significantthickness, two of the curves, C₁, C₂, are located on an inner cylinderwith radius R_(in) and two of the curves C₃, C₄, are positioned on anouter cylinder with radius R_(out). R_(in) and R_(out) define the innerand outer radii of the conductive layer. In this example, R_(in)corresponds to the aperture radius, R. See FIG. 21 which provides apartial view of the tube in cross section, showing two adjacent groovespaces G_(S) with a remaining strip S positioned between the groovespaces.

According to one method for modeling the magnetic fields, the geometryof the four curves C₁, C₂, C₃, C₄, can be determined by subdividing thehelical-shaped groove G, cut into the conductive cylinder, intoindividual elliptical-shaped groove turns, each having a center path,labeled Turn-1 to Turn-k for the given example. The space curve alongthe center line of the path for each of these curves is obtainable withEquations 2 or Equations 3. Assuming that the router bit provides acircular cutting shape of diameter D_(router) with a correspondingradius R_(router), the strip corner curves are defined by:

Strip-1: —left edge: Turn-1+R_(router)

-   -   —right edge: Turn-2−R_(router)

Strip-2: —left edge: Turn-2+R_(router)

-   -   —right edge: Turn-3−R_(router)

. . .

Strip-n: —left edge: Turn-k+R_(router)

-   -   —right edge: Turn-k+1−R_(router)

It is important to note, the corner curves C₁-C₄ of the conductivestrips are not determined by imparting a constant shift of plus or minus½W_(G) (=R_(router)) along the X-axis of the router bit curve. Thefollowing procedure outlines a process for calculating points on each ofthe corner curve space paths. It is noted that with similar procedures,space paths can be calculated for other or additional curves within theconductor strip S to vary the accuracy of the model. The displacement ofpoints relative to individual points along the center of the tool pathcurve (Equations 2 or Equations 3), to provide the corner curve paths,is determined as now described.

The slope angle at any point along the tool path curve in the unrolledview is given by the following derivative obtained from Equations 2 orEquations 3 and assuming a dipole field:

$\begin{matrix}{\frac{\mathbb{d}X}{\mathbb{d}u} = {{\tan(\alpha)} = {\frac{h}{2 \cdot \pi \cdot R} + {\frac{A_{1}}{R} \cdot {\cos\left( \frac{u}{R} \right)}}}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$FromEquation 7 one obtains the slope angle α as a function of u or θ:

$\begin{matrix}\begin{matrix}{{\alpha(0)} = {\tan^{- 1}\left( {\frac{h}{2 \cdot \pi \cdot R} + {\frac{A_{1}}{R} \cdot {\cos\left( \frac{u}{R} \right)}}} \right)}} \\{= {\tan^{- 1}\left( {\frac{h}{2 \cdot \pi \cdot R} + {\frac{A_{1}}{R} \cdot {\cos(0)}}} \right)}}\end{matrix} & {{Equation}\mspace{14mu} 8}\end{matrix}$

The resulting displacement in X-direction with a router radius ofR_(router) for any point along the tool path curve is the given by:

$\begin{matrix}{{\Delta\;{X(\theta)}} = \frac{R_{router}}{\sin(\alpha)}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

The field calculations of the coil are based on the four corner curvesC₁-C₄ that define the conductive strips. The tool path is approximatedby closely spaced points along the tool path curve. Each of these pointsis then shifted to the right or left by ±ΔX(θ) to obtain thecorresponding point on the strip corner curve. Applying thesuperposition principle for magnetic fields the Biot-Savart Law is thenused to calculate the field resulting from each of the four corner stripcurves.

The unique geometry of the conductive strips in DH coils leads to anoverall resistance of the coil that cannot be achieved with conventionalcoil geometries. Since DH coils also offer highly efficient cooling ofthe conductor, normal conducting DH coils can achieve fields that arenot possible with conventional coil windings.

With in-situ “machining” to define the conductor, the use of conductormaterials, which would be impossible with conventional windingtechniques, becomes feasible. In particular, high temperaturesuperconductors, which are brittle, can be applied to provide coils withunprecedented performance.

An exemplary coil design based on the double helix technology is shownin FIG. 22 which shows a coil consisting of two concentric cylinders. Atboth ends the machined groove departs from the coil row pattern,continuing without interruption in an axial direction toward an end ofthe aluminum cylinder. See segments “Line In-b” or “Line Out-a” of FIG.20.

As also described in FIG. 20, additional groove segments, “Line In-a”,“Line Out-b”, are then machined, one at each end, to complete thepattern. These additional groove segments “Line In-a”, “Line Out-b” eachrun alongside one of the groove segments “Line In-b” or “Line Out-a”from the end of the aluminum cylinder until they meet the coil pattern.The combination of segments “Line In-a, Line In-b” and “Line Out-a, LineOut-b” complete the formation of a “Lead-in connector (for bringingcurrent into the coil row) and a Lead-out connector for taking currentout from the coil row, e.g., to another coil row that has been machinedin a concentrically positioned cylinder.

For many applications several concentric cylinders are necessary for aDH coil to generate the required field configuration. The example ofFIG. 22 illustrates two concentric cylinders, but the number of tubes isnot limited and concentric tubes having different diameters but the sameradius of curvature can be so formed. The Figure shows the Lead-in andLead-out connectors associated with the two cylinders, configured sothat the connectors can be coupled together. In order to form acontinuous winding pattern out of two or more coil rows among concentrictubes (straight or curved), the Lead-in and Lead-out connectorsextending out from the coil patterns are interconnected. For example, asmall piece of conductive material (e.g., a conductive spacer) may besoldered between the lead connectors of the two cylinders at one end tomake the current connection as shown in FIG. 22. The two connectors atthe other end of the tube pair then form the input and output leads forthe two-layer coil.

As noted, the processes illustrated in FIGS. 15A-15C are applicable to avariety of coil row designs, including embodiments incorporating theDirect Helix design. For manufacturing processes such as illustrated inFIG. 13, the machining process can minimize manufacturing error andfurther correction resulting from manufacturing tolerances can be offsetby measuring multifield components during manufacture and incorporatingoffsetting modulations in subsequent coil rows. However, to the extent acoil row design, such a Direct Helix design includes systematic,non-random errors, the process of applying corrections to outer coilscan offset undesired multifield components generated by inner coil rows.

Although example Direct Helix embodiments have been described, numerousother designs and methods of manufacture are contemplated. For example,the aforedescribed cylinders in which helical grooves are formed mayhave an outer insulative surface (such as an anodization, a depositedcoating or other material) under which the conductive layer resides. Theinsulative surface may be formed prior to or after the groove is formedin the shape.

FIG. 23 illustrates a magnet assembly 300 formed along a central axis302, comprising an exemplary combination of interconnected coilassemblies according to the invention. In this example multiple straightsections (320, 355, 370 and 400) are designed to provide a main fieldquadrupole component and multiple curvilinear segments (330, 340, 380and 390) are configured to provide dipole main fields to bend a chargedparticle beam according to the shape of each respective segment. Theillustrated curvilinear segments may be combined function magnetswherein a dipole field is superimposed on a main dipole field. An endsegment coil assembly 310, includes a flared aperture region 312 incombination with a straight portion 320 integrally formed in the samecoil rows. The assembly 300 is connected to the section 330 having acurved shape. The section 330 is connected to the section 340 alsohaving a curved shape. The section 350 comprises a central flaredsection 360 positioned between two adjoining straight sections 355 and370. The section 350 is interposed between the section 340 and thesection 380 also having curved shape. Still another section 390 having acurved shape is interposed between the section 380 and a straight endsection 400. The section 400 has an aperture region of constant diameterand has been designed according to the optimization process of FIG. 19.In numerous embodiments the straight sections may be designed togenerate quadrupole fields (for focusing) or fields of higher order (forbeam optical corrections), and the sections having curved shapes maygenerate pure dipole fields alone or in combination with higher orderfields. The individual sections may comprise single helix or doublehelix coil rows.

Further, the combined function capabilities may be imparted bygenerating multiple modulation components in individual coil rows or byforming, e.g., concentrically about one another, a series of coil rowsthat each generate main field components of different orders. Thestraight sections, e.g., 355, 370, may alternately be drift regions ofvariable cross sections. The flared section 360 is essentially thegeometry of two flared regions 48 positioned end-to-end forming aballoon-like profile suitable for imparting, for example, high qualityfocusing fields. That is, as described for the flared regions 48, thefield quality in the flared region section 360 at a given distance Rfrom the central axis 302 exhibits is relatively high (e.g., purequadrupole) compared to straight sections of smaller (e.g., 50 percent)aperture diameter. Alternately, any of the straight or flared regionsections could be drift regions (having no fields generated therein).

Another consideration when constructing complex geometries for magnetassemblies is the continual need for focusing the charged particle beam.Typically this has been accomplished by assembling straight sectionsthat generate quadrupole fields for focusing the beam with otherfunctions such as dipole fields suitable for bending. A feature of theinvention is integration of these functions which in the past haverequired multiple sections for focusing in combination with sections forbeam bending. Conventional quadrupole magnets have a field vectorpointing in one direction over the whole length of the magnet. Such thatwhen the quadrupole field is applied to focus a beam along a horizontaldirection, it simultaneously defocuses the beam in a vertical direction.Similarly, when the quadrupole field focuses the beam along the verticaldirection, it simultaneously defocuses the beam in the horizontaldirection. For beam optical applications, where net focusing is neededin all directions it has therefore been necessary to use pairs ofconventional quadrupole magnets to achieve net focusing in alldirections.

Another feature of the invention is construction of a magnet capable ofgenerating, alone or in combination with other multipole orders, aquadrupole field that focuses in all directions. Using the double andsingle helix concepts described herein, a coil design can be constructedwhich, for example, rotates the quadrupole field about the central axisas a function of position along the axis. As a result, over the lengthof the magnet axis, the magnet focuses in all directions. This can beachieved by “twisting” the straight quadrupole coil, as described byEquation 10, around the central coil axis which extends in the Xdirection.

$\begin{matrix}{{{X(\theta)} = {{\frac{h}{2 \cdot \pi} \cdot \theta} + {\sum\limits_{n = 1}^{m\;\max}{A_{2} \cdot {\sin\left( {2 \cdot \theta} \right)}}}}}{{Y(\theta)} = {R \cdot {\cos(\theta)}}}{{Z(\theta)} = {R \cdot {\sin(\theta)}}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

“Twisting” in this context means that a transformation of the pointsdescribing the conductor path in the coil is performed that rotates eachpoint of the pattern in the Y-Z plane by an angle Φ_(twist) which isproportional to the distance of the point from the origin X=0. Thistransformation is given by the following equations.Φ_(twist) =X*Δφwith Δφ a twist angle in degree per mm advance of XX′=XRotation by the angle Φ_(twist) is given by X′ and Y′:Y′=Y*cos(Φ_(twist))+Z*sin(Φ_(twist))andZ′=−Y*sin(Φ_(twist))+Z*cos(Φ_(twist))For example, if a winding pattern has a total length of 360 mm and thetotal desired twist is 360 degrees, the required twist angle Δφ per mmis one degree. The resulting winding pattern generates a quadrupolefield with a field vector which continuously changes direction along theaxis of the magnet. Such a field will simultaneously focus the beam inall directions. Nonlinear functions are also contemplated. That is, thetwist angle per mm may vary as a function of angle or as a function ofposition along the axis. Such a nonuniform twist rate may be useful inperforming corrections or modifying the beam shape.

While the invention has been described with reference to particularembodiments, it will be understood by those skilled in the art thatvarious changes may be made and equivalents may be substituted forelements thereof without departing from the scope of the invention. Forexample, although coils have been shown to be symmetric about a straightor curved axis, numerous ones of the disclosed features can beadvantageously applied in other applications such as wherein the axis isgenerally asymmetric. The scope of the invention is only limited by theclaims which follow.

1. A method for manufacture of a conductor assembly comprising multiplecoil rows about an axis wherein outer coil rows are formed about innercoil rows, the assembly being of the type which, when conductingcurrent, generates a magnetic field or in which, in the presence of achanging magnetic field, a voltage is induced, the method comprising:forming one or more first coil rows; determining one or more deviationsfrom specifications associated with the formed one or more first coilrows, the one or more deviations each corresponding to a magnitude of amultipole field component which departs from a field specification;based on the one or more deviations, generating one or more wiringpatterns for one or more second coil rows to be formed about the one ormore first coil rows so that the magnitude of each multipole fieldcomponent that departs from the field specification is offset; andforming the one or more second coil rows in the assembly.
 2. The methodof claim 1 wherein determination of the one or more deviations is madeby calculating multipole content of the one or more first coil rows whengenerating a main field component, the method including generating anobjective function for specifying a pattern in the one or more secondcoil rows, the objective function used to proportion a magnitude of oneor more factors used to generate a modulation component in the one ormore second coil rows to offset the magnitude of each multipole fieldcomponent that departs from the field specification.
 3. The method ofclaim 2 wherein the factors are relative amplitudes and phase componentsassociated with each modulation component used to offset a multipolefield component magnitude.
 4. The method of claim 1 wherein aftergenerating the one or more wiring patterns for the one or more secondcoil rows, generation of fields by the second one or more coil rowscompensates for manufacturing errors in the one or more first coil rows.5. The method of claim 1 wherein: the assembly includes an apertureregion extending a distance R from the axis; during operation thecompleted assembly generates a main field component and at least onemultipole component for which a deviation from one of the specificationshas been offset; and along a circle in a plane transverse to the axisand having a radius at least 0.8 R from the axis, the at least onemultipole field component has a magnitude at least 10⁻⁴ less than thatof the main field component.
 6. The method of claim 5 wherein the circlehas a radius less than R.
 7. The method of claim 5 wherein along thecircle the at least one multipole field component has a magnitude atleast 10⁻⁵ less than that of the main field component.
 8. The method ofclaim 7 wherein the circle has a radius less than R.
 9. The method ofclaim 1 wherein: multiple deviations from the specifications aredetermined; and multiple modulations are included in the one or morewiring patterns so that the magnitude of each multipole field componentthat departs from the field specification is offset.
 10. The method ofclaim 1 wherein: the deviation of said multipole order from themultipole field specification is determinable relative to the mainmagnetic field, the assembly includes an aperture region extending adistance R from the axis; during operation the completed assemblygenerates the main field component and at least one multipole componentcorresponding to said multipole order; and along a circle in a planetransverse to the axis and having a radius at least 0.8 R from the axis,the at least one multipole field component has a magnitude at least 10⁻⁴less than that of the main field component.
 11. The method of claim 10wherein the circle has a radius less than R.
 12. The method of claim 11wherein along the circle the at least one multipole field component hasa magnitude at least 10⁻⁵ less than that of the main field component.13. The method of claim 12 wherein the circle has a radius less than R.